Senior year thesis
Short Description
Final year project at Monash University using OpenFOAM to simulate flapping wing aerodynamics using moving mesh techniqu...
Description
Monash University
FYP 2010
Pavaman Bilgi 19334915
MAE4902: Final Year Project Report Submission Submitted July 2010
Numerical Simulations of Flapping (Pitching) Wing Aerodynamics Pavaman Bilgi
Supervised by:
Professor Julio Julio Soria (LTRA&C) Brendon Anderson (DSTO)
Monash University
FYP 2010
Pavaman Bilgi 19334915
Summary Flapping wing aerodynamics is the basis of flight for the large majority of airborne creatures – particularly insects. It has been shown that flapping wing flow cannot be elucidated from the viewpoint of fixed wing aerodynamics. In fact flapping wing flow involves a complex interaction of fluid vortices and wing kinematics which enable things like insects perform otherwise difficult maneuvers such as rapid turning and hovering. If the mechanics of the flapping wing can be understood thoroughly, the principles may be applied to aircraft such as Micro Air Vehicles (MAVs) which are used by organizations such as the military. Computational Fluid Dynamics (CFD) and Experimental flow visualization are the primary ways of investigating these flows. Simulation is normally done in the low Reynolds number regime which is typical for insect flight and under these conditions the viscous forces in the fluid are dominant, causing vortices to roll up from the leading and trailing edges. Various aspects have been shown to affect this phenomena such as the flapping frequency, flapping amplitude angle, wing geometry and forward flight speed. Literature suggests that insects may use an optimal configuration of these parameters when flying. This report presents the results of a CFD approach to the analysis of a two-dimensional flapping wing using OpenFOAM, a computational tool using the Finite Volume Method to computationally solve the governing equations of incompressible fluid flow. This is done by comparing numerical simulations to experimental Particle Image Velocimetry (PIV) flow visualizations of flow over a quasi-elliptical wing section to establish the validity of the solver used. Subsequently the parameters of motion are varied, such as the Reynolds number, Strouhal number and the pitch angle amplitude. The way in which each of these parameters affects the development of the flow is investigated in addition to the effect of the flow on the aerodynamic forces on the wing. Results have shown that the simulated predictions are in excellent agreement with PIV visualizations. Qualitatively, all features of the flow identified in experiments have been observed in the computations. Quantitatively, the vortex size and intensity is also predicted well with a clear match being shown between simulation and experiment. The variation of Reynolds number, Strouhal number and pitch angle amplitude were all shown to have an influence on the flow evolution. The Reynolds number elicited the least effect at least in the range that was tested (500 to 1500), changing the aerodynamic wing forces only marginally. Strouhal number had a marked effect on the flow and results indicate that higher Strouhal numbers lead to higher peak values of lift and significantly more thrust production. Finally, varying the pitch angle amplitude clearly demonstrated an optimal amplitude at which lift and drag production was maximized. These findings have provided insight into the different aspects unsteady flows over a flapping quasi-elliptical wing that require consideration in terms of providing a better understanding of insect flight and the design of MAVs. Additionally, this work raises many questions for potential further study.
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Monash University
FYP 2010
Pavaman Bilgi 19334915
Nomenclature Variable
∇ ∞
Definition
Units
Fluid density
kg m3
Time
s
Velocity vector
m s
Gradient operator
⁄
⁄
m
−1
Stress tensor Specific energy Heat flux vector Kinematic viscosity
⁄
J kg
⁄ Ns⁄m W s
2
Identity matrix
⁄
Dynamic viscosity
m2 s
Fluid pressure
N m2
Free-stream velocity Frequency Reference length
⁄ m⁄s 1⁄s m
Strouhal number Reynolds number A scalar field variable Volume of control volume
m3
A vector field variable Courant Number Coefficient of cell skewness Pitch angle amplitude
degrees
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Monash University
() , , ∩
FYP 2010
Pavaman Bilgi 19334915
Chord length
m
Period of pitching motion
s
Pitch angle function
degrees
Lift and drag coefficients respectively Lift and Drag forces respectively
N
Surface area
m2
ℎ master patch face ℎ slave patch face Number of master patch faces Number of slave patch faces
ℎ master patch face to ℎ slave patch face
Weight factor of
Intersection areas of master patch face and slave patch face Radius of inlet boundary
m
-coordinate of outlet boundary
m
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Monash University
FYP 2010
Pavaman Bilgi 19334915
Table of Contents Summary.................................................................................................................................................................... Summary.................................................................................................................................................................... i Nomenclature ........................................................................................................................................................... ii List of figures and tables .......................................................................................................................................... vi 1.
2.
Introduction Introdu ction ...................................... ................... ....................................... ....................................... ....................................... ....................................... ...................................... .................................. ............... 1 1.1
Motivation Motivatio n ...................................... ................... ....................................... ....................................... ...................................... ....................................... ....................................... ............................ ......... 1
1.2
Flapping wing flow ...................................... .................. ....................................... ....................................... ....................................... ...................................... .................................... ................. 3
1.3
OpenFOAM ..................................... ................. ....................................... ...................................... ....................................... ....................................... ....................................... ............................. ......... 4
1.4
Objectives Objectiv es and outline ..................................... .................. ....................................... ....................................... ...................................... ....................................... ............................... ........... 5
Methodology Methodolo gy ...................................... ................... ....................................... ....................................... ....................................... ....................................... ...................................... ................................ ............. 6 2.1 2.1.1
The governing governin g equations ..................................... ................. ....................................... ....................................... ....................................... ...................................... ................... 6
2.1.2
Discretisation Discret isation ...................................... .................. ....................................... ...................................... ....................................... ....................................... .................................... ................. 7
2.1.3
Generating Generati ng a solution grid ....................................... ................... ....................................... ...................................... ....................................... ................................... ............... 9
2.1.4
Solving the equations ..................................... .................. ...................................... ....................................... ....................................... ...................................... ...................... ... 10
2.2
Problem definition and set-up................................................................................................................ set-up ................................................................................................................ 11
2.3
Meshing and mesh movement ..................................... ................. ....................................... ...................................... ....................................... .................................... ................ 14
2.3.1
Moving Mesh ...................................... ................... ...................................... ....................................... ....................................... ....................................... .................................. .............. 14
2.3.2
2D meshing of a symmetric symmetri c wing .................................... ................. ....................................... ....................................... ....................................... ........................ .... 17
2.4
3.
Numerical Numerica l Solution Solutio n Method ..................................... ................. ....................................... ...................................... ....................................... ....................................... ...................... ... 6
Boundary conditions conditio ns and validation validatio n ..................................... .................. ....................................... ....................................... ....................................... ........................... ....... 20
2.4.1
Setup and boundary boundar y conditions conditio ns ...................................... ................... ....................................... ....................................... ....................................... ........................ .... 20
2.4.2
Validation Validati on ...................................... ................... ...................................... ....................................... ....................................... ....................................... ....................................... ..................... 22
Results and Discussion.................................................................................................................................... Discussion.................................................................................................................................... 24 3.1
Mesh resolution, temporal resolution and domain independence study ................. ......................... ................. .................. ........... .. 24
3.2
Comparison Comparis on of results with experiment experime nt ..................................... .................. ....................................... ....................................... ....................................... ...................... 27
3.2.1
Qualitative Qualitati ve comparison compar ison ..................................... ................. ....................................... ...................................... ....................................... ....................................... ................... 27
3.2.2
Quantitative Quantita tive comparison compari son ..................................... .................. ....................................... ....................................... ...................................... .................................... ................. 33
3.3
Effect of Reynolds number .................................... ................. ....................................... ....................................... ....................................... ....................................... ...................... ... 35
3.4
Effect of Pitch angle amplitude amplitud e ...................................... ................... ....................................... ....................................... ...................................... ................................. .............. 36
3.5
Effect of Strouhal Strouh al number ..................................... .................. ....................................... ....................................... ....................................... ....................................... ...................... ... 38
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3.6 4.
FYP 2010
Pavaman Bilgi 19334915
Limitations Limitati ons ...................................... ................... ...................................... ....................................... ....................................... ....................................... ....................................... .......................... ....... 39
Conclusions Conclusi ons and Recommendations Recommend ations ...................................... ................... ....................................... ....................................... ....................................... ................................. ............. 41 Acknowledgements ............................................................................................................................................ 42
5.
References ..................................... .................. ....................................... ....................................... ...................................... ....................................... ....................................... ................................... ................ 43
Appendix A – Full Results ....................................................................................................................................... 45 Appendix B – Code Details ...................................................................................................................................... 53
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Pavaman Bilgi 19334915
List of figures and tables Figure 1 Aircraft mass vs flight regime ..................................................................................................................... 1 Figure 2 Vortices created by an insect on water ...................................................................................................... 2 Figure 3 A simulation showing the complex structure of a leading edge vortex ..................................................... 3 Figure 4 Arbitrary polyhedral volume with notation convention ............................................................................ 7 Figure 5 Structured (left) vs Unstructured (right) grids............................................................................................ grids ............................................................................................ 9 Figure 6 Non-orthogonality and skewness in two cells.......................................................................................... cells .......................................................................................... 10 Figure 7 Algorithm flow chart of code used in the present study.......................................................................... study.......................................................................... 11 Figure 8 Wing model used in DPIV experiments .................................................................................................... 12 Figure 9 Schematic of problem setup..................................................................................................................... setup ..................................................................................................................... 12 Figure 10 Example mesh setup for immersed boundary method for blades in a mixer........................................ mixer ........................................ 14 Figure 11 Example mesh deformation during motion of a rectangle in a box ....................................................... 15 Figure 12 Example GGI setup for a rotating box in a cylindrical domain ............................................................... 16 Figure 13 Meshing around an elliptical wing cross-section ................................................................................... 18 Figure 14 Meshing around a quasi-elliptical wing cross-section (with sharp edges) ............................................. 18 Figure 15 Complete inner mesh around quasi-elliptical wing section ................................................................... 19 Figure 16 Outer mesh ............................................................................................................................................. 20 Figure 17 OpenFOAM case file structure ............................................................................................................... 21 Figure 18 GGI Validation test schematic ................................................................................................................ 22 Figure 19 Subtraction of solution with GGI from that with no GGI. Maximum error is no more than 1%............ 1%............ 23 Figure 20 Peak vorticity decay over time of a single shed vortex using different discretisation schemes for the convection term ..................................................................................................................................................... 23 Figure 21 Regions of the inner computational domain.......................................................................................... domain .......................................................................................... 24 Figure 22 Boundary layer mesh resolution ............................................................................................................ 25 Figure 23 Peak lift over time for each domain independence test number .......................................................... 26 Figure 24 Domain size dimensions ......................................................................................................................... 26 Figure 25 Variation of lift and drag over time for
= , = . and = ° ...................................... 27 U
Figure 26 Numerical (left) and DPI V Experimental Data (right) sequential vorticity comparison for Re=500,
= ° ............................................................................................................................................. 28 Figure 27 Numerical (left) and DPI V Experimental Data (right) sequential vorticity comparison for Re=500, St=0.3 and = ° ............................................................................................................................................. 29 Figure 28 Numerical (left) and DPI V Experimental Data (right) sequential vorticity comparison for Re=1500, St=0.2 and = ............................................................................................................................................... 31 Figure 29 Numerical (left) and DPIV Experimental Data (right) sequential vorticity comparison for Re=1500, St=0.2 and = ............................................................................................................................................... 32 Figure 30 Vortex movement comparison with experiment for = , = . , = ° ....................... 33 Figure 31 Peak vortex intensity decay as a function of positionin domain ............................................................ 34 Figure 32 Vortex movement comparison with experiment for = , = . , = ° ..................... ................... .. 34 St=0.3 and
U
U
U
U
U
U
Figure 33 Vortex intensity as a function of vortex position ................................................................................... 35
Figure 34 Lift and drag coefficients as a function of pitch angle for a single period of motion............................. motion............................. 36 vi
Monash University
FYP 2010
Pavaman Bilgi 19334915
Figure 35 Drag coefficient as a function of pitch angle for different pitch angle amplitudes ............................... 37 Figure 36 Lift coefficient as a function of pitch angle for different pitch angle amplitudes .................................. 37 Figure 37 Lift coefficient as a function of pitch angle for different Strouhal numbers .......................................... 38 Figure 38 Drag coefficient as a function of pitch angle for different Strouhal numbers ....................................... 39 Figure 39 Complete mesh of the domain ............................................................................................................... 45 Figure 40 Vorticity solutions at intervals of
= for different (a) test number 1 (b) test number 4 and
(c) test number 6 from table 2 ............................................................................................................................... 45
Figure 41 Continuation of figure 41 ....................................................................................................................... 46 Figure 42 Vorticity solutions at intervals of
= for different (a) test number 2 (b) test number 8 and
(c) test number 9 from table 2 ............................................................................................................................... 47
Figure 43 Continuation of figure 42 ....................................................................................................................... 48 Figure 44 Vorticity solutions at intervals of
= for different (a) test number 1 (b) test number 2 and
(c) test number 3 from table 2 ............................................................................................................................... 49
Figure 45 Continuation of figure 44 ....................................................................................................................... 50 Figure 46 Vorticity solutions at intervals of
= (a) test number 5 and (b) test number 7 from table 2 .. 51
Figure 47 Continuation of figure 46 ....................................................................................................................... 52
Table 1 Experimental parameters tested ............................................................................................................... 13 Table 2 Parameters tested in the present study .................................................................................................... 13 Table 3 Boundary conditions .................................................................................................................................. 21 Table 4 Domain independence tests ...................................................................................................................... 26 Table 5 Test parameters ......................................................................................................................................... 45
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1.
FYP 2010
Pavaman Bilgi 19334915
Introduction
1.1 Motivation Although man-made flying vehicles are well known to have begun from the lauded achievement of the Wright brothers in 1903, Nature’s own flying vehicles have arguably been in existence since the creation of the Earth. While birds are assumed to be the evolutionary result of feathered dinosaurs, it is generally accepted that insects took to flight far earlier still. Despite their ubiquity however, the aerodynamics of insect flight has repeatedly continued to evade the efforts of conventional wing theory to explain the mechanisms of lift generation due to the flapping motion of their wings. Until only recently however (due in most part to advancements in experimental and numerical methods and technologies), the flow physics of flapping wings is slowly being unraveled by continued experimental flow visualization and numerical computation (Sane, 2003). While the endeavor to unveil the intricacies of insect flight is an interesting one, its salient value is in its applicability to modern aircraft design. Though the mechanisms and aerodynamics of fixed wing aircraft are generally thought to be well understood, the same cannot be said of their micro counterparts otherwise known as Micro Air Vehicles (MAVs). The aerodynamics of MAVs (and small UAVs for that matter) is defined in a different flow category to regular sized manned aircraft. Due to the combination of the relatively low velocities and length scales involved, the flight regime of these aircraft in terms of the Reynolds number becomes quite small. Refer to figure 1 (Mueller & DeLaurier, 2003) for a representation of the MAV flight regime relative to mainstream aircraft. We can see from this figure that the flight regime of MAVs coincides with that of insect flight in the low Reynolds number regime, implying that MAV design is quite amenable to the results obtained by analyzing the mechanisms of insect flight – namely the flapping motion of wings.
Figure 1 Aircraft mass vs flight regime
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Monash University
FYP 2010
Pavaman Bilgi 19334915
In addition to the above argument, there are other reasons for MAV design to adapt to the design rationale of the insect. Using current fixed wing design elements, MAVs face serious limitations in flight compared to their insect counterparts. Due to their small size, advanced and rapid control mechanisms must be employed to deal with gust response and unsteady aerodynamic loads (Rojratsirikul, Wang, & Gursul, 2009). They are unable to perform tight maneuvers as required when deployed by armed forces. Finally, they face difficulty in maintaining position without the ability to hover in one position – all limitations which flapping wing design can overcome.
Figure 2 Vortices created by an insect on water
What is so interesting and complicated about the flow over flapping wings? Flapping wings induce complex vortical structures generated from the leading and trailing edges of the wings which then interact with other features of the flow in ways that are not entirely understood. This affects the forces and performance characteristics in both hovering and forward flight. This topic has been the subject matter of research efforts in both the experimental side (Ellington, van den Berg, Willmott, & Thomas, 1996), (Weish-Fogh & Jensen, 1956) and the computational side (Wang, 2008), (Sun & Tang, 2002) by leading investigators in fluid mechanics. Both approaches have their advantages and disadvantages yet the experimental approach is limited in ways that the numerical one easily overcomes. Apparatus can often be quite expensive if not to purchase then to manufacture such as flow visualization equipment and water tunnels. More importantly however, due to the moving wing, flow visualization techniques such as digital PIV are hampered due to shadow effects and the fact that force field data (integration of surface pressures) is not easily obtained. In Computational Fluid Dynamics all information of the flow field is immediately available and provided that the parameters of the simulation are rigorously justified it can be a valuable and efficient tool in solving the problem of flapping wings. Thus, for the above reasons, a general investigation on the simulation of a flapping wing was undertaken in the low Reynolds number regime i.e.
Ο(100~2000).
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Monash University
FYP 2010
Pavaman Bilgi 19334915
1.2 Flapping wing flow Before embarking on such an investigation it is prudent to educate ourselves on the difficulties and points of interest that modeling a pitching wing presents – these are the vortex generation physics, the influence of particular flow conditions used and the differences between two and three dimensional results. Vortex generation
Vortex formation is a common phenomenon in aerodynamics. They occur due to roll up of shear layers in the flow characterized by large viscous forces (which is the case at the low Reynolds numbers considered) in situations like flow at the wing tips of large aircraft or at the hub of a wind turbine. They can have quite detrimental effects due to the localized changes in pressure that they carry with them which in turn generate fluctuating forces which can cause vibration induced damage to the structure of the flow object. However this has also been seen to be a beneficial attribute to vortex generating flows in the case of insects and birds – this is primarily due to the formation of the leading edge vortex of their flapping wings.
Figure 3 A simulation showing the complex structure of a leading edge vortex
The significance of vortex shedding over the surfaces of a wing has been highlighted in the past (Maxworthy, 1979), (Ellington, van den Berg, Willmott, & Thomas, 1996). Particularly, it is the presence of the vortex created at the leading edge of the flapping wing that makes way for lift forces much higher than those predicted by conventional wing theory. It has also been shown (in Ellington et. al.) that this vortex is essentially threedimensional and that its evolution is highly sensitive to the advance rate of the wing (Reynolds number) and the especially the wing kinematics. Indeed, a direct relationship has been observed between the roll up of the shear layers producing the leading edge vortex to the wing motion characteristics (Lentink & Dickinson, 2009) and it has also been shown that an optimization can be achieved in terms of the thrust efficiency and force coefficients of a flapping wing in relation to the kinematic parameters of the motion (Wang Z. J., 2000). This emphasizes the need to further investigate the details of the influences of the wing kinematics to the flight performance of vehicles that can be designed with such wings.
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Monash University
FYP 2010
Pavaman Bilgi 19334915
The parameter space
Various wing kinematic parameters have been tested in the literature and most of them have been shown to have pointed effects on the forces generated. Harmonic translational motion has been researched numerically with small and large amplitudes of oscillation (Lentink & Gerritsma, 2003) while frequency was varied also (Wang Z. J., 2000). It was found that by selecting certain values, the lift forces can be maximized. More studies in the same vein were performed on heaving wings (Lewin & Haj-Hariri, 2003). Again, optimal oscillatory frequencies and pitch amplitudes were proven to exist and the variation of aerodynamic efficiency was established. The geometry of the actual wing has also been investigated (Lentink & Gerritsma, 2003) while fixing other motion parameters like frequency and pitch amplitude. The conclusions from these studies were that the geometry did not significantly affect the lift forces generated when compared to the effect of minor changes in the motion parameters. Rotational parameters have also been varied (Dickinson M. H., 1994) and the results of this research have shown that the axis of rotation and the rotation speed profile are heavily important in the lift generation of the wing. Given the above broad area of literature pertaining to a wide spectrum of motion parameters of flapping wings, the scope of this project has been chosen to be confined to a focused analysis of a few parameters including the Reynolds number, geometry and pitch frequency. In this way it is hoped that this project will make a more valuable contribution to the field than the ‘shotgun’ approach of detailing results for various independent groups of parameter values. Is there a necessity to simulate 3-dimensional flows?
An ongoing question posed by researchers in the problem of flapping wings is the accuracy of two-dimensional simulations and experimental results compared to the three-dimensional ones. Additional issues introduced in the three-dimensional problem are unsteady flow mechanisms, wing flexibility and wing tip vortex generation. Studies have been made into the comparison between two and three dimensions (Wang Z. J., 2004) where three dimensional experimental results of the Robofly (experimental robotic test-bed for flapping insect wings at UC Berkely) to two dimensional results. The findings there and elsewhere (Dong, Mittal, Bozkurttas, & Najjar, 2005) (Blondeaux, Fornarelli, & Guglielmini, 2005) confirm that the two dimensional studies often over-predict the lift and drag forces and do not account for the energy loss that occurs in the third dimension. Three dimensional vortices were observed to exist in the wake of finite span wings. Nevertheless, studies have been performed in recent times on two-dimensional flapping wing flow (Bos, Lentink, & van Oudheusden, 2008) (Thaweewat, Bos, van Oudheusden, & Bijl, 2009). The justification used in these cases is that while there are unresolved flow features in the 3D case, there is sufficient similarity between the 3D and 2D cases (Wang Z. J., 2000b) to ensure that a parametric study in 2D can be quite useful. Given that a 3D computation can be quite expensive, the scope of this project has until now been limited to the two dimensional simulation of a solely pitching wing.
1.3 OpenFOAM 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 4
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Pavaman Bilgi 19334915
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.
1.4 Objectives and outline While the proposed two-dimensional simulations of a pitching wing are intended to give insight into the effects of different motion and flow parameters, the extent of the reliability of these results needs to be established first for a given set of parameters. This is because it is known that differences between numerical and experimental results vary when varying the parameter values (Wang Z. J., 2004). To this end it was determined that experimental results should be procured in order to both provide an indication of the applicability of the numerical results and to give some measure of validation of the code used. Additionally, due to OpenFOAM being a relatively new software in CFD arena compared to mainstay packages such as Fluent, CFX and StarCD, and more so due to its open-source nature, it would be of benefit to document the method of setting up this type of problem in terms t erms of solvers and utilities used. Thus, the objectives of the project are stated below. 1. To gain experience in and document the operation of OpenFOAM in relation to moving-mesh problems. 2. To determine the quality of 2D simulations of a pitching wing in OpenFOAM by verifying with experimental results. 3. To extend the experimental results by establishing the effects of varying the parameters of motion and wing geometry on the flow field and forces generated. To address these objectives, the next few sections deal with the methodology used to setup and numerically solve the flow field around a pitching wing. This involves a description of the Finite Volume discretisation method used by OpenFOAM and basic overview of the solution algorithm used. Further, the method used to handle the moving boundary is covered followed by the rationale for the grid construction and boundary conditions used. Finally, the motion parameters and important variables are defined and the experimental results used for comparison are briefly c overed. Subsequently a presentation of the results is given. This entails a comparison of simulation data to experimental data and a qualitative and quantitative discussion of the various points of similarity and difference. The next part will detail the results of varying the motion and flow parameters. It will be seen that these change in parameters affect the development and size of the leading edge vortex which consequently has ramifications on the forces generated. Finally, the conclusions and recommendations for future work will follow.
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Pavaman Bilgi 19334915
2. Methodology
2.1 Numerical Solution Method The Finite Difference Method, Finite Element Method (FEM) and the Finite Volume Method (FVM), are the conventional approaches to solving the governing partial differential equations (PDEs) of fluid flow, or any three dimensional continuum mechanics problem in general in both commercial and non-commercial software packages. The FVM is normally used for Computational Fluid Dynamics problems and this is the case with OpenFOAM® as stated in earlier sections. This section will cover the governing equations, the discretisation method, grid generation and a brief discussion on the solution algorithm.
2.1.1
The governing equations
The governing equations describing the viscous fluid flow are the Navier-Stokes equations. These are a coupled set of non-linear PDEs. One may derive these by consideration of the mass, momentum and energy conservation laws (Anderson, 1991). Refer to t he literature for further information.
Mass cons. Momentum Cons. Energy cons. Stress tensor
+ ⋅ () = 0 () + ⋅ () = ⋅
() + ⋅ () = ⋅ () − ⋅ + = − + ⋅ ⋅ I + ( + )
(2.1)
(2.2)
(2.3)
2 3
Here, the stress tensor is defined for a Newtonian fluid. Since the present work deals with flow around insect wings, it is considered to be incompressible and more importantly, laminar (Williamson, 1995). Given these conditions, we may make some simplifications to the above equations. When a fluid is incompressible, the flow velocity is nowhere greater than 30% of the local speed of sound and thermal expansion effects may be neglected. Under these conditions, the Navier-Stokes equations reduce to, Mass cons. Momentum Cons.
⋅ =0 + ⋅ () = − + 2
(2.4) (2.5)
This system of equations is closed so there is no need for further relations such as the energy equation.
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Pavaman Bilgi 19334915
For the purposes of comparison of different solutions of the incompressible NS equations it is preferred that the dimensionless variables be used. These are defined below:
∗ = ∕ ∞ , ∗ = ⋅ , ∗ = ⁄, ∗ = ⁄� ⋅ ∞ , ∗ = ⁄ 2
(2.6)
We may substitute equation (2.6) into (2.4) and (2.5) to obtain the following non-dimensional form of the continuity and momentum equations: Mass cons. Momentum Cons.
∗=0 ⋅ ∗ + ⋅ (∗∗) = −∗ + 1 ∗
(2.7)
2
Here, two new dimensionless variables are introduced known as the Strouhal number, (
(2.8)
= ⁄ )
= ∞ ⁄). 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 (). and the Reynolds number (
2.1.2
Discretisation
To solve the equations (2.7) and (2.8) in a numerical fashion, the solution domain needs to be discretised. In this particular case this means dividing the domain into a set of discrete control volumes that are joined by faces and do not overlap. Each field variable is the solved at the center of each control volume. Logically, the smaller the volumes become, the more accurate the solution will be and as the volume approaches zero, the error of the solution also approaches zero.
Figure 4 Arbitrary polyhedral volume with notation convention
The center of each control volume,
is defined by the equation, ∫ ( − ) = 0. Each of these control
volumes (tetrahedral, hexahedral or polyhedral) then has a certain number of faces and points associated with it and the governing equations are to be approximated over these cells. This is done by assuming a linear variation of any field variable across any two cells centers. Thus, using the notation in figure 4 where
and 7
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Pavaman Bilgi 19334915
denote the center of and normal vector of each face respectively, we can use a Taylor expansion on any scalar field variable,
Here the
about the cell center of any volume to obtain, () = + ( − ) ⋅ (∇) + �| − | () = �( + Δ) − () + (Δ) Δ
(2.9) (2.10)
terms represent the order of the truncation error. Using this formulation, we can discretise the
incompressible NS equations. We start from equation (2.5) and note that in velocity, there are two spatial terms to discretise and one temporal term. First we represent equation (2.5) in integral form:
+ ⋅ () − ⋅ () =
(2.11)
Before discretising each term we establish the discretisation of a general volume, surface, divergence and gradient integrals of a scalar field variable, vector, •
•
•
(this may represent a velocity component for example) and
(perhaps a velocity vector). ∫ () = ∫ + ( − )() = ∫ + ∫ ( − ) ⋅ ( ) ≈ ∫ ⋅ = ⋅ ∫ ⋅ = ∮ ⋅ = ∑ ∫ ⋅ ≈ ∑ ⋅
In the above, Gauss’ formula is utilized which allows one to express a volume integral as a surface integral on the volume. The integral is then evaluated using the finite number of faces on the volume in question.
∫ = ∮ = ∑ ∫ ≈ ∑ Finally, the terms evaluated at the surface, must be interpolated using a scheme of the users’ choice. The •
surface interpolation schemes used for this project were the linear, the Koren and Van Leer limiter schemes. The details of these schemes are not within the scope of this project although it was found that the Van Leer scheme provided the best results. We can now discretise the momentum equation appropriately. Convection term
⋅ () = ⋅ () = ⋅ () =
Here,
is used to denote the mass flux, given by = ⋅ ()
Diffusion term
⋅ () = ⋅ () = (⋅)
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Monash University
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Pavaman Bilgi 19334915
Temporal term
Here, a number of different time differencing schemes can be adopted. For this project, the backward difference second order implicit scheme was used (Hirsch, 1 988).
= 1 3 − + 1 − 2 Δ 2 In this formulation, denotes the state of the field variable at time . As an aside, from stability and error +1
1
propagation analysis, the Courant number defined as,
= Δ⋅ Δ must be less than unity for the solution to be stable over time.
2.1.3
Generating a solution grid
As stated in the previous section, the computational domain of the solution must be divided into a finite amount of cells. It turns out that one may do this in a structured or unstructured manner. Figure 5 shows examples of both structured and unstructured grids. As can be seen, using structured grids is more elegant but more importantly it makes the cell ordering a lot more organized. A numerical solver may use this fact to improve the efficiency with which a solution is computed. Additionally, it is easier to ensure near perfect orthogonality of cells in a structured grid over an unstructured grid.
Figure 5 Structured (left) vs Unstructured (right) grids
The limitation of the structured grid arises in the difficulty of creating it around complicated geometries which is a case for using an unstructured grid. The cell shape and number of faces may be varied in an unstructured grid which permits one to maintain high degrees of cell orthogonality even around relatively complex boundaries. Compared to a structured grid however, the degree of cell orthogonality achievable of the unstructured grid is not as high. Cell orthogonality or more generally, cell ‘quality’ is crucial to the accuracy of a numerical solution. This is because it affects the interpolation of the field variables between cell-centers at the cell surfaces. A representation of non-orthogonality and skewness is given in figure 6. Here, it is the angle
that is the
measure of orthogonality and this must be as low as possible for an accurate flow solution and particularly, to
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minimize the error of the diffusion term. Skewness is increased when the line connecting two cell centers moves away from the center of the interface of those two cells. The quantitative measure of cell skewness is given by = | | | |.
⁄
Figure 6 Non-orthogonality and skewness in two cells
For this project it was decided to use structured grids wherever possible in the vicinity of the wing and the near wake region and use unstructured grids a sufficient distance away from the wing. This is so that the important flow features near the wing can be accurately resolved while the grid cell size can easily be inflated away from the wing where comparatively no flow structures exist.
2.1.4
Solving the equations
The above algebraic equations provide a basis for solving the incompressible NS equations. However two complications arise when solving these equations. Firstly there is the non-linear convection term in the momentum equation. One may opt to solve the equation using complex and expensive non-linear solvers or one can perform linearization on the equations and solve them iteratively. The code used for this project takes the linearization route so that the existing velocity fields are used to calculate the face fluxes in the non-linear convection term. The second difficulty is that the equations are coupled – the velocity field depends on the pressure and viceversa. To this end, the PISO scheme was used (Issa, 1986). While the detailed explanation of this scheme is not within the scope of this report, it will suffice to say that it essentially consists of three steps. First, the momentum equation is solved for velocity using the old pressure field. Then, the pressure field is updated using an equation combining the continuity and momentum equations. Finally, the velocity is also corrected using the updated pressure field using a modified version of the momentum equation.
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Thus, the solution steps can be outlined as follows: Initialise the field variables using the initial conditions provided.
Update the time to get new velocity and pressure values.
Setup the matrices for the equations using the face fluxes from the previous time step.
Go through the PISO loop and iterate until the velocity and pressure field changes converge below the tolerances.
Compute lift, drag, vorticity and any other equations to be solved. If turbulence equations included, solve them.
If it is the final time, stop. Otherwise go back to the second step. Figure 7 Algorithm flow chart of code used in the present study
2.2 Problem definition and set-up The present study is largely an extension of existing experimental results of flapping wing sections and a validation of the code used. Thus the problem was defined to replicate that of the experiment which is explained in this section. 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 intensity levels levels in the core region are purportedly less than 0.35%. 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. 11
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A quasi-elliptical cross-section aerofoil was used with a chord length of 31mm and an aspect ratio of 3.3 where
= ⁄. The
the aspect ratio is defined as the square of the span divided by the chord length of the section; diagram below illustrates this.
2
Figure 8 Wing model used in DPIV experiments
The pitching displacement profile was a sinusoidal one enforced by stepper motors which rotate the wing about the mid-chord point. The motors were programmable so that different pitch frequencies and amplitudes could be tested, allowing for different displacement profiles. The diagram below illustrates the basic scheme of the setup for a particular pitch amplitude of 20 degrees.
Figure 9 Schematic of problem setup
The flow was characterized by the Reynolds number and the Strouhal number introduced earlier. The Reynolds number is a representation of the ratio of convection time to diffusion time in a flow. Alternatively, it is the ratio of inertial to viscous forces in a fluid. Its magnitude will determine what kind of flow structures will be responsible for the forces developed on a flapping wing – for example at low Reynolds numbers (which are considered in the present study), the viscous term is dominant, leading to viscous phenomena such as shear layers and the creation of vortices. The Strouhal number is normally used in the context of vortex shedding flows and is quite relevant to flapping wing flows. It is defined by the flapping frequency multiplied by the reference length of the flapping motion divided by the reference flow velocity. 12
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In the DPIV experiments the Reynolds number and Strouhal number were varied. These are summarized in the table below. Table 1 Experimental parameters tested
500 500 1500 1500
0.3 0.4 0.2 0.4
20 20 20 20
The sinusoidal motion may be parameterized in terms of the Reynolds number and Str ouhal number as follows. The displacement follows a function of the form, (2.12) sin(2 ) () = sin( ∕ where the reference length where can be expressed in terms of the Strouhal number as, = ∞ ∕ used for the Strouhal number is the total displacement of the trailing edge during the pitching motion. This can
be expressed as,
tan( ) = ⋅ tan( Further, the free stream velocity,
∞ can be represented in terms of the Reynolds number, as, ∞ = ⋅ ⁄
Substituting this into the displacement equation (2.12) we arr ive at,
⋅ ⋅ ⋅ radian () = sin 2 ⋅ tan
(2.13)
2
where
= 1 × 10− m ⁄s which is the value for water at the temperature used in the experiment. 6
2
Thus, the period of the pitching motion then becomes,
seconds = tan ⋅ ⋅ 2
(2.14)
In order to extend the findings of the experimental results, further values of in addition to varying
and were tested in this study
, the pitch angle amplitude. This is summarized in the table below. Table 2 Parameters tested in the present study
Test #
(°)
1
2
3
4
5
6
7
8
9
500
500
500
1000
1000
1500
1500
500
500
0.2
0.3
0.4
0.2
0.4
0.2
0.4
0.3
0.3
20
20
20
20
20
20
20
30
40
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Finally, this study also aims to extend the results of the experiment by considering the lift and drag forces generated on the wing over time and analyse the effect of the vortex evolution on these forces. This will be looked at in terms of the lift and drag coefficient defined as below.
= ⁄(0.5 ⋅ ∞ ) = ⁄(0.5 ⋅ ∞ ) 2
(2.15)
2
(2.16)
2.3 Meshing and mesh movement In this section a discussion of the rationale for the meshing of the wing and domain is presented, followed by an explanation of the approaches considered to move the mesh in order to solve the problem of the moving wing and finally the approach used.
2.3.1
Moving Mesh
One of the key challenges of performing CFD calculations on insect wing dynamic flows is that of the treatment of the moving boundary problem. Since in a conventional CFD problem the computational node locations are specified in advance of the computation itself, the fact that the location of the boundary condition imposed changes in time warrants that new approaches be used. Several methods to address this problem have been implemented in the literature and these were all co nsidered before deciding on a particular method to use. Immersed Boundary Method
One way to address the problem is to impose the boundary condition of a moving boundary at different nodes for each time-step. This allows one to have as structured a grid as one wants. However this method introduces challenges since the boundary does not always adhere to the grid nodes at all times and thus the term, ‘Immersed Boundary’ (Mittal & Iaccarino, 2005). Hence, interpolation algorithms are required in order to find the appropriate boundary condition to impose at the nearest node locations. This can necessitate dense gridding in the vicinity of the boundary movement and the interpolation algorithms need to be of a high order – both of which can be computationally expensive.
Figure 10 Example mesh s etup for immersed boundary method for blades in a mixer
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Rotating Reference Frame
Another approach is to define a new reference frame in which the boundary does not move. In the case of a pitching wing for example, this would be a non-inertial reference frame that rotates with the wing. In this reference frame, since the wing does not move, the boundary condition at the wing then becomes a no-slip condition, while the Coriolis and centrifugal acceleration terms must be added to equation (2.5). These additional terms add complexity to the solving method however, and the enforcing of the far-field boundary condition also becomes complicated. This is because technically at the far-field, in a non-inertial reference frame rotating with the body, the velocity is infinite by definition of the far-field (Sun & Tang, 2001). Dynamic Meshing
The dynamic meshing method involves explicit topological changes in the mesh. When the individual nodes of the boundary are moved at each time, the solver includes a routine which computes the redistribution of the nodes of the mesh of the previous previous time-step. This method of treating moving boundaries boundaries is required by fluidstructure interaction problems since the movement of the boundary is not prescribed. The mesh movement is done in two ways – the first is to treat the connections between nodes as springs and to use for example, Laplacian smoothing to redistribute the nodes. The second is to move each node individually, otherwise known as point-by-point methods. All dynamic mesh methods require a separate mesh motion solver which can come at a heavy computational expense.
Figure 11 Example mesh de formation during motion of a rectangle in a box
General Grid Interface
The GGI method was developed for the purposes of turbo-machinery simulations where separate meshes are normally created for different components of the system such as a rotor and a stator. The main advantage is that it removes the necessity of topologically altering the mesh all the time while still letting the mesh move. 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. Due to the simplicity of implementation and the computational efficiency of the method, this method was chosen as the best way to approach the pitching wing problem.
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Figure 12 Example GGI se tup for a rotating box in a cylindrical domain
Because the GGI method operates on the interface of two meshes, another advantage is that the resolutions of the two meeting meshes need not be the same. The field variables of the flow are controlled over the interface by weighting factors which are used in the interpolation between the meshes (Beaudoin & Jasak, 2000). A GGI is represented by a meeting of a master and shadow patch. The master patch is composed of while the shadow patch is composed of
faces
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.
= ⋅
(2.17)
= ⋅
(2.18)
Equations (2.17) and (2.18) 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 neighboring faces are defined as those faces on the opposing patch which intersect the face in question. Further constraints are imposed on these weighting factors in order for the interpolation to be conservative.
= 1
(2.19)
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= 1 ⋅ = ⋅ = |∩ |
(2.20) (2.21)
Refer to the nomenclature for variable definitions. From equation (2.21) we can see that the weight factors are simply the ratio of the intersection area of a patch face with each of its neighboring patches. That is to say,
= ∩
(2.22)
∩ =
(2.23)
Given this formulation for the weighting factors, it remains only to calculate the intersection areas of faces and neighboring faces which is a geometric consideration. This involves the determination of which are the neighborhood faces for each face on the master or slave patches and efficient algorithms are implemented in the mesh motion solver to take care of this.
2.3.2
2D meshing of a symmetric wing
Due to the solely pitching motion of the wing, a circular GGI was the most appropriate to use for the problem. The GGI was placed at 2 chord lengths from the center of rotation. The meshing was approached by ensuring that maximum cell quality was achieved in the region of the domain where the vortex structures were expected before giving consideration to the near wake region. Subsequently, an unstructured approach was used to inflate the cell size toward the domain boundaries. Inner mesh
On the inside of the GGI region, the inner part of the mesh was a circular region containing the wing. Two general symmetric wing shapes were considered during this project – blunt edges and sharp edges. In the case of blunt edges the wing can be described analytically as an ellipse with a semi-major and minor axis. The grid constructed for the blunt edge wing shape is shown in figure 13. Since the circle is simply a specific case of an ellipse it is not difficult to gradually release the elliptical plan-form of a wing into a circle while maintaining a high degree of cell quality near the wing. When using sharp leading and trailing edges however the wing profile is no longer well defined and is discontinuous. It was decided to release the ‘sharp-edged’ wing into an ellipse which is then released into the outer circle. The top and bottom surfaces of the sharp-edged wing were layered with highly structured quadrilateral (hexahedral in 3D) mesh elements whereas the regions on the ends of the wing were filled with high density high quality triangular (tetrahedral in 3D) mesh elements. This arrangement conveniently resulted in an intermediate elliptical profile which was then released in a conformal fashion to the outer circle as shown in figure 15.
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Figure 13 Meshing around an elliptical wing cross-section
Figure 14 Meshing around a quasi-elliptical wing cross-section (with sharp edges)
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Figure 15 Complete inner mesh around quasi-elliptical wing section
Outer mesh
The outer mesh interfaces with the inner mesh at the GGI and also extends to the far-field boundaries where the flow is expected to approach the free stream velocity profile. The nominal cell size in each area of the outer mesh must be such that while the flow features in that area are adequately resolved, but also that excessive computational time is not wasted in those regions. Important areas to consider in the outer region are the forward and wake regions of the wing and top and bottom far-field boundary areas of the flow. The outer mesh used is pictured in figure 16. It was decided that high density quadrilateral meshing would be used for the near wake region while triangular meshing with inflation would be applied to the rest of the domain. As will be shown in later sections, this was also for the convenience of performing the grid resolution study where it became easy to adjust the resolution of the parameterized wake region while keeping the rest of the mesh unchanged. It is also important to note the choice of the ‘OH’ type grid. This is a standard approach when considering flow over a wing and is done so that the inlet conditions imposed on the left boundary do not create unwanted reflections at the corners of a normal ‘H’ type grid. 19
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Figure 16 Outer mesh
2.4 Boundary conditions and validation This section details file structure and boundary conditions of the problem setup in OpenFOAM. Subsequently the results of a brief validation study on the discretisation schemes used and the GGI method is presented.
2.4.1
Setup and boundary conditions
OpenFOAM is a command-line driven software package for UNIX systems. As such all simulation parameters are specified in specifically formatted text files in allocated directories. The directory structure of a case is pictured in figure 17. 20
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Figure 17 OpenFOAM case file structure
The constant directory contains full information of the mesh and is also where the dynamic mesh method is specified. The system directory contains information on the discretisation schemes used and the solution algorithms chosen to solve the equations for the solver to be used ( icoDyMFoam in this case, which is the standard solver for the incompressible NS equations in OpenFOAM). Finally the time directories are destinations for solution output with the exception being the 0 time directory where initial and boundary conditions are specified. Refer to Appendix B for the full details of the files. To begin solving the equations intialisation of the flow is necessary. This was done using potentialFoam, the potential flow solver in OpenFOAM to get a sensible starting distribution of velocity and pressure. The resultant flow solution from this solver is then used as initialization to the icoFoam solver. However additionally, boundary conditions must be specified in order to have a closed system of equations after discretisation described in section 2.1.2. The boundary conditions specified are summarized in table 3. Table 3 Boundary conditions
zeroGradient
Outlet
∞
zeroGradient
0
Top/Bottom
symmetryPlane
symmetryPlane
Wing surface
movingWallVelocity
zeroGradient
Inlet
The values of U and p at the boundaries must be such that their effect on the flow in the vicinity of the wing is negligible. To this end, on the top and bottom of the domain (above and below) the wing, a symmetry boundary condition is used for both U and p. This condition specifies the boundary to act like a mirror, effectively setting the normal component of the solution at that boundary to zero. The inlet and outlet boundaries are given a combination of Neumann and Dirichlet boundary conditions. At a boundary the Dirichlet type condition sets the value of a variable while a Neumann condition sets the value of the gradient of a variable. Thus the Neumann boundary condition allows for one dimensional variation on the 21
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boundary. Since pressure perturbations propagate further upstream than downstream, pressure at the inlet is given a Neumann condition while the velocity is set to free-stream (Dirichlet). Similarly, since velocity perturbations propagate much further downstream than upstream, the velocity at the outlet is given a Neumann condition while the pressure is set to free-stream (Dirichlet). Finally, the condition imposed at the boundary of the wing itself was that of a moving wall for obvious reasons. This means that this boundary condition is one that changes in time and OpenFOAM provides a routine to be able to calculate the velocity of a moving boundary at all times. Should the velocity imposed at the wing boundary not match the velocity of the boundary we should expect mass flux through the walls of the wing. An excerpt from the flux field at the wing boundary shows that the moving wall velocity calculator is working very well since we achieve (close to) zero mass flux through the wing wall boundary during motion: wingWall mass flux { 440 ( 0 -2.42338e-26 2.26182e-26 1.29247e-26 -1.29247e-26 2.58494e-26 2.58494e-26 -2.58494e-26 -5.16988e-26 …
2.4.2
Validation
Before beginning simulations a validation study was performed on the GGI method to ensure that the results using this method were satisfactory. This was done by using the case of flow over a cylinder where the vortex shedding is reminiscent of the wake of a flapping wing. Starting from flow over a stationary cylinder at a Reynolds number of 1 × 10 5 , a circular GGI was placed in the wake of the cylinder with the inner mesh region rotating sinusoidally. The solution was then compared to the case with no GGI and if the operation at the GGI is working as expected, the differences between the solutions should be minimal.
Figure 18 GGI Validation test schematic
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In figure 18 we can see a schematic of the validation test. Furthermore in figure 19 we can see that the subtraction of the solution with the GGI and that without gives a maximum error of less than 1%. This was done with a sinusoidal rotation of the inner mesh at a Strouhal number of 0.3 where the length scale used was the diameter of the cylinder. While this is a rather crude method of establishing the validity of the GGI method, due to time constraints it would have to do. It should be noted that results may be different for different rotation frequencies and Reynolds numbers. Further testing is required to establish these effects.
Figure 19 Subtraction of solution with GGI from that w ith no GGI. Maximum error is no more than 1%
The second validation study concerned the appropriate discretisation scheme for the convection term in equation (2.11). This is because it was seen that the evolution of the generated vortices was dependent on the amount of diffusion the convection scheme was producing. To do this, a set of discretisation schemes commonly used for the convection term were used and the results were measured in terms of the vortex preservation of the scheme i.e. the intensity of vortices after a specified amount of time. As can be seen in figure 20, the vanLeerV scheme provided the best vortex energy preserving properties over time as the vortices convect down-stream. This scheme was used throughout the present study. Refer to Appendix B for a full account of all discretisation schemes used in the fvSchemes file. 1.6
Peak vorticity vs time vanLeerV
y t i s n e t n I x e t r o V
Koren Linear
0 0
Time
7
Figure 20 Peak vorticity decay over time of a single shed vortex using different discretisation schemes for the convection term
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3. Results and Discussion In this section the computed solutions for the various flow and motion parameters outlined in section 2.2 are presented. This is preceded by the results of the mesh resolution, temporal resolution and domain independence studies which are used throughout the rest of the results. In particular since, as mentioned in section 1.2 we are interested in the evolution of the vortices shed by the wing, this section predominantly focuses on the vorticity field of the solution and the movement of regions of peak vorticity. Subsequently a brief analysis of the influence of these vortices on the pressure forces on the wing will be given in the form of lift and drag plots followed by discussion. This will be done for varying Reynolds and Strouhal number and Pitch amplitude (refer to table 2).
3.1
Mesh resolution, temporal resolution and domain independence study
The purpose of the resolution study is to ensure that all relevant flow features of interest are adequately resolved by the resolution of the discretisation used. Since we expect that as the resolution tends to infinity the solution tends to the exact solution, this is done by proving that r efinement (increasing the resolution) does not change the solution more than a threshold amount. For the present study, the aspect of the solution observed while increasing resolution was the vorticity field and the threshold ‘residual’ or solution change with resolution increase was 1%. The following was performed at
= 500 and = 0.3.
Mesh and temporal resolution
Figure 21 Regions of the inner computational domain
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The inner and immediate wake regions of the mesh are where the mesh resolution is critical. Hence these areas were the focus of the mesh resolution study. Due to the implementation of the GGI method described in section 2.3.1, the mesh resolutions of the inner and wake regions can be altered independently since the inner and outer regions of the mesh are topologically disconnected. Thus, a separate r esolution study was conducted for each of these regions. The resolution of the inner mesh was ascertained before that of the wake region since flow moves from the inner region to the wake region. The resolution in the vicinity of the wing and the wake region was increased by decreasing the width of the average cell in that region. Each time the resolution was increased, the vorticity solution with the new mesh was interpolated onto the old mesh and subtracted from the solution of the old mesh. If this was greater than 1% anywhere in the domain, the resolution was further increased.
− | < 0.01 for all , in the inner mesh | |
|
Figure 22 Boundary layer mesh resolution
A similar process was used for determining the appropriate temporal resolution of the simulation i.e. finding the right time step size to use. A further limitation in the size of the time-step used is in the use of the GGI method. Since a large time-step corresponds to a greater movement (rotation) of the inner mesh, this may affect the quality of the interpolation at the GGI. This resulted in a requirement of 1500 time-steps per period of pitching motion of the wing. For a Reynolds number of 500 and a Strouhal number of 0.3, this corresponds to a time-step of 1.5 milliseconds. Domain independence
Since the boundary conditions are derived at the ‘far-field’ boundaries (at a distance infinitely far away from the wing), the actual distance of these boundaries needs to be established to obtain domain independence. Due to the structure of the domain used, two parameters need to be determined; the radius of the inlet boundary,
and the -coordinate of the outlet boundary, . As and tend to infinity, we expect that 25
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the solution (if adequately resolved by the mesh) to approach the exact solution. As before for the mesh resolution, the domain size was increased until a solution residual of less than 1% was observed. In the case of domain independence however, it was the residuals of lift force that were observed. Table 4 Domain independence tests
Test 1 2 3 4
) 2 ^ m / N ( n a p s t i n U / t f i L
5 10 15 20
10 15 20 25
Peak lift vs test #
4.05E-05
3.90E-05
3.75E-05 1
2
Test #
3
4
Figure 23 Peak lift over time for each domain independence test number
Domain size independence is clearly achieved at test number 3. This is represented in figure 24.
Figure 24 Domain size dimensions
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Comparison of results with experiment
Results were taken once the solution of the computations reached periodic steady state conditions. That is to say when,
( + ) = ( ) where
represents any flow field variable. We see the flow reaching periodic steady state conditions in figure
25 of the lift and drag forces produced on the wing for test number 2 of table 2 – this is clearly around 4 to 5 periods of motion from initialization.
Lift and drag vs Pitch angle from t = 0 8.00E-06
) 2
m / N ( a e r a t i n u r e p e c -25 r o f g a r D , t f i L
4.00E-06
0.00E+00 -1 5
-5
5
15
25 Lift
-4.00E-06 Drag
-8.00E-06 Pitch angle (degrees) Figure 25 Variation of lift and drag over time for
3.2.1
= , = . and = °
Qualitative comparison
Figure 26 and 27 show the comparison of in-plane vorticity distributions for computations and experiment for
∞ = 0.16667 / and = 2.18 seconds ( = 0.46 Hz). The comparisons are made in intervals of 10 degrees of pitching motion starting from () = 0° . The spatial dimensions are scaled by the reciprocal of the chord length, 1⁄ , the velocity is scaled by the reciprocal of the free-stream velocity, 1⁄∞ and the vorticity is scaled by the chord length divided by the velocity scale, ⁄∞ . test number 2 in table 2 where
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 26 (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 26 (d). 27
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(a)
(b)
(c)
(d)
Figure 26 Numerical (left) and DPIV Experimental Data (right) sequential vorticity comparison for Re=500, St=0.3 and
= ° 28
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(a)
(b)
(c)
(d)
Figure 27 Numerical (left) and DPIV Experimental Data (right) sequential vorticity comparison for Re=500, St=0.3 and
= ° 29
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Figure 27 (a) 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 27 (c) 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 line. Thus although the flow on the bottom of the wing is not captured in the experimental visualizations visualizations due to the laser shadow cast by the wing, this region of the flow is simply the reflection of the top side about = 0 with a phase lag of half a period of motion, or /2. It should be noted that since we are dealing with symmetrical motion, the flow is symmetrical about the
Figure 28 and figure 29 depict comparisons of vorticity distributions for test number 6 in table 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 28 (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 28 (c) we see that at
= −20° the TEV has formed and detached
while the LEV is proceeding down-stream across the wing surface. As before in test number 2, we see that the shear layer is advancing across the top surface of the wing – 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 29 (a) 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. Thus, since the LEV is not absorbed, it is released at the trailing edge to join the TEV formed at time
= ⁄2
from when the LEV was formed. This produces an oblique wake in which counter rotating vortex pairs travel down-stream. 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 28 (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.
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(a)
(b)
(c)
(d)
Figure 28 Numerical (left) and DPIV Experimental Data (right) sequential vorticity comparison for Re=1500, St=0.2 and
= 31
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(a)
(b)
(c)
(d)
Figure 29 Numerical (left) and DPIV Experimental Data (right) sequential vorticity comparison for Re=1500, St=0.2 and
=
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Quantitative comparison
Since it has been established that the vortex generation and evolution is the primary point of interest in flapping wing flows, the numerical solutions were compared with experiment by using the peak vorticity of vortices generated (at the leading and trailing edges) and the location of those vortices while observing their variation over time and space. This comparison should give an indication of how well the code is predicting the formation, decay and movement of the vortices as they travel over the surface of the wing, encounter shear layers and interact with each other down-stream in the wake. We first consider the position of the LEV and TEV over time from the time of birth for test number number 2 in table 2 (the flow solution is pictured in figures 26 and 27). Figure 30 shows a favorable comparsion of the movement of the LEV and TEV over time – the simulations accurately predict the time of detachment of the vortices and their speed as they travel downstream.
Vortex position vs Time 2.5
2 ) n o i t i s o 1.5 p ( e t a n i 1 d r o o c x 0.5
LEV TEV ExpLEV ExpTEV
0 0
0.5
1
1.5
Time (s)
Figure 30 Vortex movement comparison with experiment for
2
2.5
= , = . , = °
Furthermore, in figure 31, 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 of the decay being observed later on. The quicker decay of the LEV can be 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 26 (c). Figures 32 and 33 show the peak vorticity comparisons for test number 6 in table 2. 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 2. Nonetheless patent agreement is found between simulation and experiment in the behavior of the vortices.
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Vortex intensity vs Position 35 30 ) s / 1 ( y t i c i t r o v k a e P
25 LEV 20
TEV ExpLEV
15
ExpTEV 10 5 0 0
0 .5
1
1.5
2
2.5
x-coordinate Figure 31 Peak vortex intensity decay as a function of positionin domain
Figure 32 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 29 (c) 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 29 (a) where the LEV is halfway across the upper surface of the wing. This is where the 3
Vortex position vs Time
2.5 ) e t a 2 n i d r o o c - 1.5 x ( n o i t 1 i s o P
LEV TEV ExpLEV ExpTEV
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
Time (s) Figure 32 Vortex movement comparison with experiment for
= , = . , = °
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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. As before in figure 30, figure 33 shows the variation of vortex intensity with vortex position for test number 6 of table 2. 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 32. 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 29 (a).
Vortex intensity vs position
45 40 ) s / 1 ( y t i c i t r o v k a e P
35 30 LEV 25
TEV
20
ExpLEV
15
ExpTEV
10 5 0 0
0.5
1
1.5
2
2 .5
3
Vortex position (x-coordinate) (x-coordin ate) Figure 33 Vortex intensity as a function of vortex position
3.3
Effect of Reynolds number
The effect of changing the Reynolds number was observed by comparing the solutions obtained from tests 1, 4 and 6 from table 2. In these tests, the Strouhal number and the pitch angle amplitude are held constant at
= 0.2 and = 20° respectively. In order to quantify the effect of changing the Reynolds number, the variation of the lift and drag coefficients, and with the pitch angle, was plotted and compared. The results showed that the effect of Reynolds number was minimal. Looking at the flow solutions, at
= 500, the LEV was seen to be consumed by the advancing shear layer. Increasing to 1000 results in the creation of a stronger LEV which manages to survive the journey over the surface of the wing and travel into the wake. Increasing further to 1500 magnified this effect and creates a more sparse wake structure.
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Meanwhile the TEV remains unaffected by the change in Reynolds number. Refer to figure 40 and 41 in Appendix A for vorticity solutions of these tests.
and shown in figure 34 confirm the above observations. We can see that increasing the Reynolds number marginally increases over all pitch angles and also reduces . One interesting observation however is the thrust production (indicated by negative ) increase with increasing . For = 500 thrust is produced at 5° < || < 14°. However at = 1500 the range of thrust production expands to 1° < | | <
The behavior of
16°. In addition, the thrust coefficient is almost tripled as the Reynolds number goes from 500 to 1500. This is most likely due to the survival of the LEV over the rear surface of the wing at the higher Reynolds numbers.
Lift and drag coefficients vs Pitch angle 2.5 2 1.5 1
t n e i c i f f e o c g a r -25 D / t f i L
0.5 0 -20
-15
-10
-5
0
5
10
15
20
25
-0.5 Cd500 -1 -1.5
Cl500 Cd1000 Cl1000
-2 -2.5
Cd1500 Cl1500
Pitch angle (degrees) Figure 34 Lift and drag coefficients as a function of pitch angle for a single period of motion
3.4
Effect of Pitch angle amplitude
was tested at 20°, 30° and 40° while keeping the Reynolds and Strouhal numbers constant at = 500 and = 0.3 respectively. These conditions correspond to tests 3, 8 and 9 in The pitch angle amplitude,
table 2. The frequency of pitching motion was therefore (according to equation (2.14)) 0.31, 0.29 and 0.2 Hz
respectively. The vorticity plot comparisons shown in figures 42 and 43 in Appendix A show the differences in the flow at intervals of
Δ = 0.125 or 1⁄8 ℎ of the period of pitching motion. We see that as is increased, while the
formation and travel of the TEV is relatively unaffected but that of the LEV undergoes interesting changes. The
strength of the LEV is ostensibly much greater as the pitch angle amplitude is increased however it is its 36
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movement that plays a critical role in the generation of forces on the wing. In the case of 20° and 30° the LEV remains close to the wing surface and since vortices are regions of low pressure, the LEV is responsible for the generation of lift and thrust forces as it travels over the wing. In the 40° case however we notice that the LEV is generated quite early and also separates from the leading edge early. Subsequently it is deflected by the rear of the wing during the latter part of the stroke and moves away from the wing surface. The lift and drag coefficient plots (figures 35 and 36) confirm the above observations by showing that lift and thrust generation are maximized at 30° while it begins to drop after this point (for
= 40°) due to the moving
. We also notice that the lift generated for the pitch angle amplitude of 40° is generally less than that produced at = 30°. away of the LEV at higher
Drag coefficient coefficient vs Pitch Angle 1.6 1.2
t n e i c i f f e o c g a r D -50
Cd20
0.8
Cd30 Cd40
0.4 0 -40
-30
-20
-10
0
10
20
30
40
50
-0.4
Pitch angle Figure 35 Drag coefficient as a function of pitch angle for different pitch angle amplitudes
Lift coefficient vs Pitch angle 4 3 2 1 t n e i c i f f-50 e o c t f i L
0 -40
-30
-20
-10
0
10
20
30
40
50
-1 -2 -3
Cl20 Cl30 Cl40
-4
Pitch angle Figure 36 Lift coefficient as a function of pitch angle for different pitch angle amplitudes
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Effect of Strouhal number
The effect of Strouhal number was analysed by looking the flow solutions of tests 1, 2 and 3 where the Reynolds number and pitch pitch angle amplitude are held constant at Strouhal number, Hz respectively.
= 500 and = 20° respectively. The
is tested at 0.2, 0.3 and 0.4. This corresponds to a pitch frequency of 0.31, 0.46 and 0.61
Figure 37 and 38 show the lift and drag coefficient variation with pitch angle. It is clearly evident that at the higher Strouhal number, larger, more intense vortices are created at the leading edge (stronger LEVs) in addition to stronger TEVs. This is obviously the cause of the greater production of lift and thrust than at the lower strouhal numbers (comparing
= 0.3 to 0.2 for example). We notice something different happen when
the Strouhal number moves up to 0.4 however. The lift profile changes such that much lower lift coefficients
are registered at the lower pitch angles whereas much higher lift coefficients are found at the higher pitch angles. Furthermore, looking at the drag coefficient curves we see that thrust production is markedly increased at
= 0.4 with the range of pitch angles at which thrust is generated being significantly increased also. Lift coefficient vs Pitch angle 6
4
t n e i c i f f e o c -2 5 t f i L
2
0 -20
-15
-10
-5
0
5
10
15
20
25
-2 Cl0.2 -4
Cl0.3 Cl0.4
-6
Pitch angle Figure 37 Lift coefficient as a function of pitch angle for different Strouhal numbers
Figures 44 and 45 in appendix A reveal the reasons behind the observations noted above. While indeed we do see the stronger LEVs associated with the higher pitch frequencies (higher strouhal numbers), at
= 0.4 the
frequency is so high as to not allow the LEV to convect downstream before it is destroyed in the return stroke of the wing. This means that the LEV (a region of very low pressure) remains at the leading edge at all times
thus inducing the generation of thrust. Furthermore this allows the advancing shear layer created at the rear of the wing to advance over the surface of the wing unimpeded by the LEV – this leads to very high lift coefficients being generated.
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Drag coefficient coefficient vs Pitch angle 1.6 Cd0.2
1.2 t n e i c i f f e o c g a r D -25
Cd0.3
0.8
Cd0.4 0.4 0 -20
-15
-1 0
-5
0
5
10
15
20
25
-0.4 -0.8
Pitch angle Figure 38 Drag coefficient as a function of pitch angle for different Strouhal numbers
3.6
Limitations
Before concluding it is necessary to outline the most prominent limitations that were faced in obtaining the aforementioned results in this section. These should be addressed in any future work continuing the present investigation. It is important to note firstly that the analysis done in section 3.1 concerning mesh and temporal resolution was only done for certain flow and motion conditions and that other conditions may yield different results. Ideally the conditions which impose the most stringent conditions on the mesh and time step should have been tested but due to time constraints, this was not able to be done. Because of this a single mesh was used throughout the study which may not always have resolved the flow accurately in all conditions (especially at the higher Reynolds numbers and higher pitch frequencies). Another limitation due to time constraints was the limitation to two dimensional simulations. Section 1.2 established the three dimensionality of flapping wing vortical flows. However due to the sheer computational expense of three dimensional simulations, this was not pursued. Consequently, the stretching of the vortices
and energy dissipation in the third dimension ( -axis) will not have been captured which will affect the vorticity distribution in the
-plane. There is also a limitation from the experimental point of view due to the finite
span of the wing. This will create unwanted wing tip vortices that will affect the flow along the span of the wing to some extent. Finally a glaring limitation also brought on by time constraints was the limited number of tests that could be performed. Although for example it was found that Reynolds number had a relatively minimal effect on the results the case could have been different for different pitch frequencies and pitch angle amplitudes. Ideally the parametric study would be performed on a much wider and more comprehensive range of parameter values. 39
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4. Conclusions and Recommendations Numerical solutions of the incompressible Navier-Stokes equations have been generated for the transient flow problem of the aerodynamics of a two dimensional pitching wing section using OpenFOAM®. The wing profile was that of a quasi-elliptical shape and simulations were conducted in the low Reynolds number regime of the order of 200 to 2000 and Strouhal numbers (pitch frequencies) relevant to insect locomotion. The dynamic mesh method used was the General Grid Interface method which proved to be an effective technique to approach the problem both from viewpoint of computational efficiency and ease of implementation. A key advantage of this method was its ability to handle different rotational motion parameters without any modification which other mesh motion methods require. Vadidation tests and comparison with experimental data show that questions must still be posed as to the accuracy of the method under different rotation speeds and different mesh resolutions at the i nterface. Domain and mesh resolution studies showed that a mesh of at least 500,000 elements was required for the
and directions was required for the effects of the boundary conditions imposed to be negligible. Simulations at = 500, = 0.3 and = 1500, = 0.2 with a pitch angle amplitude, = 20° were compared to experimental flow visualizations using problem and a domain of at least 15 chord lengths in both the
Digital PIV equipment at the LTRA&C. Vorticity fields showed that the 2D simulations accurately predicted creation of both the leading and trailing edge vorticies including the evolution of those vortices with time. Thus, although the flow is in principle three dimensional, the present study shows that two dimensional simulations can potentially deliver accurate results. The degree of accuracy seemed to vary between test conditions, however and this is a point requiring further experimentation and study. The results of the PIV experiments were extended to include the effects of Reynolds number, Strouhal number and pitch amplitude angle. The variation of the Strouhal number and pitch angle amplitude induced interesting effects on the development and evolution of the leading edge vortex while the Reynolds number had little effect (in the range that was tested). It was found that when the Strouhal number was increased for a fixed Reynolds number and pitch amplitude, the strength of the LEV was significantly increased leading to increasing thrust and lift generation. Increasing the pitch amplitude angle for fixed Reynolds and Strouhal numbers on the other hand also increased vortex strength but also altered vortex movement resulting in the observation of an optimal pitch angle amplitude in terms of lift and drag. The above conclusions are a summary of an introductory study into flapping wing flows using OpenFOAM®. Many complexities of the flow have been identified and no doubt will attract further study in this area. Based on the findings the following questions for further a nalysis arise: 1. Increasing the distance of the GGI from the wing reduces the error incurred at the GGI but reduces the maximum allowable time-step for computation. Thus, what is the optimum distance of the GGI from the wing? 2. The present study has been performed for a quasi-elliptical cross section with sharp edges. How do the conclusions change (if at all) for differing wing geometries?
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Acknowledgements 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 Computing Infrastructure without which the present work would not have been possible.
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5. References Anderson, J. (1991). Fundamentals of Aerodynamics, second edn. McGraw-Hill Inc. Beaudoin, M., & Jasak, H. (2000). Turbomachinery section. Retrieved 2010, from University of Zagreb, Power Engineering Department: http://powerlab.fsb.hr/pe http://powerlab.fsb.hr/ped/kturbo/OpenFOAM/Berlin2008/SessionIV/ d/kturbo/OpenFOAM/Berlin2008/SessionIV/ Blondeaux, P., Fornarelli, F., & Guglielmini, L. (2005). Numerical Experiments on flapping foils mimicking fish like locomotion. Physics of Fluids , 113601-12. Bos, F. M., Lentink, D., & van Oudheusden, B. W. (2008). Influence of wing kinematics on aerodynamic performance in hovering insect flight. Journal of Fluid Mech. , 341-368. Dickinson, M. H. (2000). How animals move: an integrated view. Science , 100-106. Dickinson, M. H. (1994). The effects of wing rotation on unsteady aerodynamic performance at low Reynolds number. The Journal of Experimental Biology , 179-206. Dickinson, M. H., & Gotz, K. G. (1993). Unseady aerodynamic performance of model wings at low Reynolds numbers. The Journal of Experimental Biology , 45-64. Dickinson, M. (1999). Wing rotation and the aerodynamic basis of insect flight. Science , 1954-1960. Dickson, W. B., & Dickinson, M. H. (2004). The effect of advance ratio of the aerodynamics of revolving wings. Journal of Experimental Biology , 4269-4281.
Dong, H., Mittal, R., Bozkurttas, M., & Najjar, F. (2005). Wake structure and performance of finite aspect-ratio flapping foils . 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno . Ellington, C. P., van den Berg, C., Willmott, A. P., & Thomas, A. (1996). Leading-edge vortices in insect flight. Nature , 626-630.
Green, M., Parker, K., & Soria, J. (2005). 2D DPIV of a Pitching Aerofoil. Fourth Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion .
Hirsch, C. (1988). Numerical Computation of internal and External Flows, Volume I: Fundamentals of Numerical Discretisation. John Wiley & Sons.
Issa, R. I. (1986). Solution of the implicitly discretised fluid flow equation by operator splitting. Journal of Computational Physics , 2053-2059.
Jasak, H. (2010). List of Papers: Hrvoje Jasak . Retrieved July 1st, 2010, from Hrv's Homepage: http://www.h.jasak.dial.pipex.com/ Lentink, D., & Dickinson, M. H. (2009). Rotational accelerations stabilise leading edge vortices on revolving fly wings. Journal of Experimental Biology , 2705-2719. 43
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Lentink, M., & Gerritsma, M. H. (2003). 43rd AIAA Fluid Dynamics Conference and Exhibit. Orlando . Lewin, G. C., & Haj-Hariri, H. (2003). Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. Journal of Fluid Mech. , 339-362. Maxworthy, T. (1979). Experiments on the weis-fogh mechanism of lift generation by insects in hovering flight. part 1. dynamics of the 'fling'. Journal of Fluid Mechanics , 47-63. Mittal, R., & Iaccarino, G. (2005). Immersed Boundary Methods. Ann. Rev. Fluid Mech. , 239-261. Mueller, T. J., & DeLaurier, J. D. (2003). Aerodynamics of Small Vehicles. Ann. Rev. Fluid Mech. , 89-111. Pedro, G., Suleman, A., & Djilali, N. (2003). A Numerical study of the propulsive efficiency of a flapping hydrofoil. International Journal for Numerical Methods in Fluids , 493-526. Rojratsirikul, P., Wang, Z., & Gursul, I. (2009). Unsteady fluid-structure interations of membrane airfoils at low Reynolds numbers. Experiments in Fluids , 859-872. Sane, S. P. (2003). The aerodynamics of insect flight. The Journal of Experimental Biology , 4191-4208. Sun, M., & Tang, J. (2001). Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. The Journal of Experimental Biology , 55-70.
Sun, M., & Tang, J. (2002). Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. The Journal of Experimental Biology , 55-70.
Thaweewat, N., Bos, F. M., van Oudheusden, B. W., & Bijl, H. (2009). Numerical study of vortex-wake interactions and performance of a two-dimensional flapping foil. 47th AIAA Aerospace Sciences Meeting, Orlando .
Wang, Z. J. (2008). Aerodynamic efficiency of flapping flight: analysis of a two-stroke model. Journal of Experimental Biology , 234-238.
Wang, Z. J. (2000b). Two dimensional mechanism for insect hovering. Physical Review Letters , 2216-2219. Wang, Z. J. (2004). Unsteady forces and flows in low Reynolds number hovering flight:two-dimensional computations vs robotic wing experiments. The Journal of Experimental Biology , 449-460. Wang, Z. J. (2000). Vortex shedding and frequency selection in flapping flight. Journal of Fluid Mech. , 323-341. Weish-Fogh, T., & Jensen, M. (1956). Biology and physics of locust flight. i. basic principles of insect flight: a critical review. Phil. Trans. Royal Society London , 415-458. Williamson, C. (1995). Fluid Vortices (Vol. Vol. 30). Kluwer Academic Publishers.
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Appendix A – Full Results This section contains flow visualizations of vorticity for all test conditions shown in table 2. All plots have the same vorticity scale. Further details of the tests are provided in table 5 below. Table 5 Test parameters
Re
St
500
0.3
500
(deg) (secs)
∞ (m/s)
Write interval (s)
Frequency (Hz)
0.0167
0.091
0.46
20
2.184
1.456E-03
0.4
20
1.638
1.092E-03
0.0167
0.068
0.61
500
0.2
20
3.276
2.184E-03
0.0167
0.136
0.31
500
0.3
30
3.464
2.309E-03
0.0167
0.144
0.29
500
0.3
40
5.035
3.356E-03
0.0167
0.210
0.20
1000
0.2
20
1.638
1.092E-03
0.0333
0.068
0.61
1000
0.4
20
0.819
5.460E-04
0.0333
0.034
1.22
1500
0.2
20
1.092
7.279E-04
0.0500
0.045
0.92
1500
0.4
20
0.546
3.640E-04
0.0500
0.023
1.83
Figure 39 Complete mesh of the domain
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(a)
(b)
(c)
Figure 40 Vorticity solutions at intervals of
= ⁄ for different (a) test number 1 (b) test number 4 and (c) test number 6 from table 2 45
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(a)
(b)
(c)
Figure 41 Continuation of figure 41
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(a)
(b)
(c)
Figure 42 Vorticity solutions at intervals of
= ⁄ for different (a) test number 2 (b) test number 8 and (c) test number 9 from table 2 47
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(a)
(b)
(c)
Figure 43 Continuation of figure 42
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(a)
(b)
(c)
Figure 44 Vorticity solutions at intervals of
= ⁄ for different (a) test number 1 (b) test number 2 and (c) test number 3 from table 2 49
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(a)
(b)
(c)
Figure 45 Continuation of figure 44
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(a)
(b)
Figure 46 Vorticity solutions at intervals of
= ⁄ (a) test number 5 and (b) test number 7 from table 2
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(a)
(b)
Figure 46 Vorticity solutions at intervals of
= ⁄ (a) test number 5 and (b) test number 7 from table 2
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(a)
(b)
Figure 47 Continuation of figure 46
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(a)
(b)
Figure 47 Continuation of figure 46
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Appendix B – Code Details The computations performed in the present study were made possible by the computing time shares granted by the Australian National University (ANU) at the National Computational Infrastructure (NCI). A 64-bit release of OpenFOAM-1.5-dev was compiled under a Linux operating system. This section contains the fvSchemes and fvSolution files contained in the system directory of the case directory. These files detail the discretisation schemes
and solvers used, respectively. Subsequently the modified mixerGgiFvMesh.C file is appended which was required to facilitate the sinusoidal motion of the wing.
fvSchemes /*--------------------------------*- C++ -*----------------------------------*\ | ========= | | | \\ / F ield | OpenFOAM: The Open Source CFD Toolbox | | \\ / O peration | Version: 1.3 | | \\ / A nd | Web: http://www.openfoam.org |
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Appendix B – Code Details The computations performed in the present study were made possible by the computing time shares granted by the Australian National University (ANU) at the National Computational Infrastructure (NCI). A 64-bit release of OpenFOAM-1.5-dev was compiled under a Linux operating system. This section contains the fvSchemes and fvSolution files contained in the system directory of the case directory. These files detail the discretisation schemes
and solvers used, respectively. Subsequently the modified mixerGgiFvMesh.C file is appended which was required to facilitate the sinusoidal motion of the wing.
fvSchemes /*--------------------------------*- C++ -*----------------------------------*\ | ========= | | | \\ / F ield | OpenFOAM: The Open Source CFD Toolbox | | \\ / O peration | Version: 1.3 | | \\ / A nd | Web: http://www.openfoam.org | | \\/ M anipulation | | \*---------------------------------------------------------------------------*/ FoamFile { version format class object }
2.0; ascii; dictionary; fvSchemes;
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // ddtSchemes { default CrankNicholson 0.5; } gradSchemes { default grad(p) } divSchemes { default div(phi,U) }
Gauss linear; Gauss linear;
none; Gauss vanLeerV;
laplacianSchemes { default none; laplacian(nu,U) Gauss linear corrected; laplacian(rAU,pcorr) Gauss linear corrected; laplacian(rAU,p) Gauss linear corrected;
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laplacian((1|A(U)),p) Gauss linear corrected; } interpolationSchemes { default linear; interpolate(HbyA) linear; interpolate(1|A) linear; } snGradSchemes { default } fluxRequired { default pcorr; p; }
corrected;
no;
// ************************************************************************* //
fvSolution /*--------------------------------*- C++ -*----------------------------------*\ | ========= | | | \\ / F ield | OpenFOAM: The Open Source CFD Toolbox | | \\ / O peration | Version: 1.3 | | \\ / A nd | Web: http://www.openfoam.org | | \\/ M anipulation | | \*---------------------------------------------------------------------------*/ FoamFile { version format class object }
2.0; ascii; dictionary; fvSolution;
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // solvers { pcorr PCG { preconditioner tolerance relTol
DIC; 1e-07; 0;
}; p PCG { preconditioner
DIC;
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1e-06; 0.05;
}; pFinal PCG { preconditioner tolerance relTol }; U PBiCG { preconditioner tolerance relTol };
DIC; 1e-07; 0;
DILU; 1e-05; 0;
} PISO { nCorrectors 4; nNonOrthogonalCorrectors 1; pRefCell 0; pRefValue 0; }
// ************************************************************************* //
Notes: •
PCG: Pre-conditioned Conjugate Gradient solver. It is the iterative solver for solving the pressure equation.
•
PBiCG: Pre-conditioned Bi-stab conjugate Gradient solver. It is the solver used to solve the pressure-
velocity coupling. •
DIC: Diagonal Incomplete Choleski (a pre-conditioner)
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DILU: Diagonal Incomplete LU decomposition pre-conditioner
mixerGgiFvMesh The new variables introduced are declared like so in the mixerGgiFvMesh.C file: Pi_ = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628 03482534211706798214808651328230664709384460955058223172535940812848111745028410270193 85211055596446229489549303819644288109756659334461284756482337867831652712019091456485 66923460348610454326648213393607260249141273724587006606315588174881520920962829254091 71536436789259036001133053054882046652138414695194151160943305727036575959; Re_ = readScalar(dict_.lookup("Re")); c_ = readScalar(dict_.lookup("c")); theta_ = readScalar(dict_.lookup("theta")); St_ = readScalar(dict_.lookup("St")); nu_ = readScalar(dict_.lookup("nu"));
55
Monash University
FYP 2010
Pavaman Bilgi 19334915
theta_ = theta_*Pi_/180; Info
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