Analysis of non symmetrical flapping airfoils and their configurations

ANALYSIS OF NON-SYMMETRICAL FLAPPING AIRFOILS AND THEIR CONFIGURATIONS TAY WEE BENG (B. Eng. (Hons.), M. Eng., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Summary The objective of this project is to improve the performance of the efficiency, thrust and lift of flapping wing. The first phase of study investigates the effects of different flapping parameters (reduced frequency, Strouhal number, pitch amplitude and phase angle) and the airfoil’s shape on its efficiency, average thrust and lift coefficients (h, Ct and Cl ). Interactions between the parameters are also studied using the Design of Experiment (DOE) methodology. The next phase of the research aims to investigate the effect of active chordwise flexing. A total of five flapping configurations are selected and the objective is to see if flexing can help to further improve these cases. Moreover, the effect of center of flexure, leading/trailing edge flexing, a form of single-sided flexing and the use of non-symmetrical airfoil are also investigated. The last phase of the research investigates the effect of the arrangement of the airfoils in tandem on the performance of the airfoils by varying the phase difference and distance between the two airfoils. Results from the DOE show that both the variables and shape of the airfoil have a profound effect on the h, Ct and Cl . By using non-symmetrical airfoils, average lift coefficient as high as 2.23 can be obtained. The average thrust coefficient and efficiency also reach high values of 2.53 and 0.61 respectively. The Cl is highly dependent on the airfoil’s shape while Ct is influenced more heavily by the variables. Efficiency falls somewhere in between. Two-factor interactions are found to exist among the variables. This shows that it is not sufficient to analyze each variable individually. The chordwise flexing results show that flexing is not necessarily beneficial for the performance of the airfoils. However, with the correct parameters, efficiency is as high as 0.76 Analysis of Non-symmetrical Airfoils and their Configurations i by placing the flexing center at the trailing edge. Average thrust coefficient is more than twice as high from 1.63 to 3.57 if flapping and flexing occur under the right conditions. Moreover, the single-sided flexing also gives an average lift coefficient as high as 4.61 for the S1020 airfoil. The shape of the airfoil does alter the effect of flexing too. It has also been found that in non-optimized flapping configuration, flexing is more likely to improve the efficiency of the airfoil. For the tandem airfoil arrangement simulations, all the different flapping configurations show improvement in the h, Ct or Cl when the distance between the two airfoils and the phase angle between the heaving positions of the two airfoils are optimal. The average thrust coefficient of the tandem arrangement managed to attain more than twice that of the single one (4.84 vs. 2.05). On the other hand, the average lift coefficient of the tandem arrangement also increased to 4.59, as compared to the original single airfoil value of 3.04. The research data obtained from the studies of DOE, airfoil’s flexing and tandem configuration will enable the design of a better performing ornithopter in terms of efficiency, thrust and lift production. Analysis of Non-symmetrical Airfoils and their Configurations ii Acknowledgements The author wishes to express sincere appreciation of the assistance and suggestions given by the Supervisor, Assoc. Prof Lim Kah Bin The author would also like to thank post doctorate research fellow Dr. Hu Yu and Masters student Mr Nguyen Duc for their ideas and contribution. The author wishes to thank Dr. Tsai Her Mann for a brief but rewarding advice. The author is grateful to the technologists Mrs Ooi, Ms Tshin, Mr Zhang, Ms Hamida, Mdm Liaw and Mrs Too in Control Lab and 2, for providing excellent computing facilities to carry out the project. Furthermore, the author wishes to thank the systems engineers Mr. Wang Junhong and Neo Eng Hee for the technical help they have provided at Supercomputers and Visualisation Unit (SVU). Lastly, the author would like to thank his family members and friends who have given him many useful suggestions and moral support. Analysis of Non-symmetrical Airfoils and their Configurations iii Table of Contents Summary . i Acknowledgements . iii Table of Contents iv List of Figures . ixii List of Tables xii Nomenclature xiv Introduction . Literature Review 2.1 Solvers for flapping wing simulation . 2.2 Kinematics of Flapping Configuration . 2.3 Airfoil flexing 13 2.4 Biplane/Tandem Airfoil Arrangement 15 Code Development and Validation . 18 3.1 Unsteady Lattice Vortex Method (UVLM) . 18 3.1.1 Code Development Summary 18 3.1.2 Theory of the UVLM . 19 3.1.2.1 Basic formulation 19 3.1.2.2 Defining the kinematics of the wing 22 3.1.2.3 The wake shedding and roll-up procedure 24 3.1.2.4 The influence coefficients 25 3.1.2.5 The linear set of equations of Newman boundary condition . 26 3.1.2.6 Pressure, velocity and load computations . 26 3.1.2.7 Code implementation 27 3.1.3 Modifications and improvements to code . 28 Analysis of Non-symmetrical Airfoils and their Configurations iv 3.1.3.1 Geometry model 28 3.1.3.2 Movement of the wing 29 3.1.3.3 Graphic user interface and vortex visualization 29 3.1.3.4 Vortex blob modifications 30 3.1.4 3.2 Verification of the UVLM 34 Structured Collated Navier Stokes Solver (SCNSS) . 38 3.2.1 Algorithm of the SCNSS . 38 3.2.1.1 Fractional step method 38 3.2.1.2 C-grid and grid motion 41 3.2.1.3 Boundary conditions . 44 3.2.1.4 Force coefficients and efficiency computation . 44 3.2.2 Verification of the SCNSS . 45 3.2.3 Grid Convergence Test 51 3.2.3.1 Quantitative validation – Cl and Ct measurements 51 3.2.3.2 Qualitative validation – Vorticity Diagram . 53 3.3 Staggered Cartesian Grid Navier Stokes Solver with Immersed Boundary 56 3.3.1 Algorithm of the IBCNSS 57 3.3.1.1 Fractional step method 57 3.3.1.2 Cartesian grid and boundary conditions . 60 3.3.1.3 Force coefficients and efficiency . 60 3.3.2 Verification of the IBCNSS . 61 3.3.3 Grid Convergence Test 63 3.3.3.1 Quantitative validation – Cl and Ct measurements 63 3.3.3.2 Qualitative validation – Vorticity Diagram . 64 3.3.3.3 Parallelizing of the IBCNSS Code . 65 Methodology in Experimental study . 67 4.1 Design of Experiment (DOE): Box-Behnken (BB) Design . 67 4.2 Airfoil Active Chordwise Flexing 72 Analysis of Non-symmetrical Airfoils and their Configurations v 4.3 Results and Discussions from the DOE 84 5.1 The Box-Behnken (BB) Test 86 5.2 Significance and Effect of Variables on Efficiency . 88 5.3 5.4 5.5 Tandem Airfoils . 76 5.2.1 Significance of k and q0 and their Interaction . 89 5.2.2 Significance of St 91 5.2.3 Significance of f and q0 . 93 5.2.4 Comparison of Efficiency of Different Airfoils 94 Significance and Effect of Variables on Thrust . 95 5.3.1 Significance of k and q0 and their Interaction . 97 5.3.2 Significance of k and f and their Interaction 98 5.3.3 Significance of St and its Interactions with f and q0 . 100 5.3.4 Interaction between q0 and f 103 5.3.5 Comparison of Thrust of Different Airfoils 104 Significance and Effect of Variables on Lift . 104 5.4.1 Reduced Frequency k 106 5.4.2 Significance of f . 107 5.4.3 Two-factor Interactions . 108 5.4.4 Comparison of Lift of Different Airfoils 108 Chapter Summary 109 Results and Discussion for Airfoil Chordwise Flexing .110 6.1 6.2 Flexing – Pure Heaving 110 6.1.1 Double sided flexing (Figure 6.1) .110 6.1.2 Single-sided flexing (Figure 6.4) 115 Flexing – ME Configuration (h0 = 0.75, k = 0.2, q0 = 30o, f = 90o) 119 6.2.1 Double sided flexing (Figure 6.7) .119 6.2.2 ME Single-sided flexing (Figure 6.9) . 122 Analysis of Non-symmetrical Airfoils and their Configurations vi 6.3 ME (20o) Configuration (h0 = 0.75, k = 0.2, q0 = 20o, f = 90o) 124 6.3.1 6.3.1.1 6.4 6.5 Double sided flexing (Figure 6.10) 124 Single-sided flexing (Figure 6.13) . 129 Flexing – MT Configuration (h0 = 0.42, k = 0.6, q0 = 17.5o, f = 120o) 131 6.4.1 Double sided flexing (Figure 6.14) 131 6.4.2 Single-sided flexing (Figure 6.15) . 133 Flexing – ML Configuration (h0 = 0.15, k = 1.0, q0 = 17.5o, f = 120o) 135 6.5.1 Double sided flexing (Figure 6.16) 135 6.5.2 Single-sided flexing (Figure 6.17) . 137 6.6 Comparison of Effect of Flexure between Different Flapping Configurations . 139 6.7 Comparison of Effect of Flexure between Different Airfoils under Similar Flapping Configurations . 140 6.8 Chapter Summary 141 Results and Discussion for Tandem Airfoils . 144 7.1 7.2 7.3 Tandem ME Configuration 144 7.1.1 f12 = -90o, 1.25 ≤ d12 ≤ 2.50 . 144 7.1.2 d12 = 2.0, -180o ≤ f12 ≤ 150o . 149 Tandem MT Configuration 151 7.2.1 f12 = -90o, 1.25 ≤ d12 ≤ 2.50 . 151 7.2.2 d12 = 1.75, -180o ≤ f12 ≤ 150o . 153 Tandem ML Configuration . 155 7.3.1 f12 = -90o, 1.25 ≤ d12 ≤ 2.50 . 155 7.3.2 d12 = 2.0, -180o ≤ f12 ≤ 150o . 157 7.4 Effect of d12 and f12 on the Performance of the Airfoils 158 7.5 Chapter Summary 159 Applying Simulation Results to Actual Ornithopters . 161 Conclusion . 162 Analysis of Non-symmetrical Airfoils and their Configurations vii 10 Recommendations 165 11 References . 167 12 Publication from this Research . 174 12.1 Journal Articles (In Review) 174 12.2 Conference Papers . 174 Appendices 175 A DOE 175 A.1 Test Configurations and Results for BB test . 175 B Chordwise Flexing . 182 B.1 C Results for Chordwise Flexing . 182 Instructions to Execute Codes 212 C.1 UVLM User Instructions . 212 C.2 SCNSS User Instructions . 213 C.2.1 Compilation 213 C.2.2 Execution 214 C.2.2.1 Without Morphing . 214 C.2.2.2 With Morphing . 215 C.2.3 Output . 216 C.2.3.1 Without Morphing . 218 C.2.3.2 With Morphing . 218 C.3 IBCNSS User Instructions . 219 C.3.1 Compilation 219 C.3.2 Execution 219 C.3.2.1 For Airfoil 220 C.3.2.2 For Airfoils in Tandem . 220 C.3.3 Output . 221 Analysis of Non-symmetrical Airfoils and their Configurations viii Appendices 0.2 SD ND 0.20 SD ND 1.42 SD ND 2.37 0.3 ND ND SD ND ND SD ND ND SD xfc = 1.0 af h Ct Cl NACA0012 NACA6302 S1020 NACA0012 NACA6302 S1020 NACA0012 NACA6302 S1020 -0.4 0.14 ND SD 1.41 ND SD 3.62 ND SD -0.3 0.14 ND 0.14 1.40 ND 1.52 3.49 ND 4.61 -0.2 0.18 ND 0.20 1.64 ND 2.05 2.50 ND 4.11 -0.1 0.24 SD 0.24 2.22 SD 2.37 1.72 SD 2.58 0.0 0.25 0.21 0.26 2.50 2.10 2.33 0.41 0.30 1.35 0.1 0.24 ND ND 2.27 ND ND -1.22 ND ND 0.2 0.18 ND ND 1.69 ND ND -2.48 ND ND Analysis of Non-symmetrical Airfoils and their Configurations 208 Appendices Table B.14: ML configuration results for ≤ a f ≤ 0.4 xfc = 0.0 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 0.1 0.18 0.15 2.47 2.12 1.23 0.98 0.2 0.11 0.10 2.53 2.36 -0.33 1.89 0.3 SD 0.11 SD 3.57 SD 1.65 0.4 ND 0.04 SD 1.92 SD 2.31 xfc = 0.5 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 0.1 0.17 0.17 1.95 1.95 1.91 1.91 0.2 SD SD SD SD SD SD xfc = 1.0 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 0.1 SD 0.17 SD 1.40 SD 1.52 0.2 ND 0.13 ND 1.04 ND 1.12 Analysis of Non-symmetrical Airfoils and their Configurations 209 Appendices Table B.15: ML configuration results for -0.4 ≤ a f ≤ xfc = 0.0 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 -0.1 0.23 0.26 1.16 1.22 0.86 1.93 -0.2 0.19 0.26 0.50 0.74 0.17 1.62 -0.3 0.05 0.18 0.11 0.39 0.10 1.29 -0.4 ND 0.05 ND 0.13 ND 1.81 xfc = 0.5 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 -0.1 0.25 0.23 1.43 1.31 0.33 1.28 -0.2 SD 0.22 SD 0.67 SD 0.41 xfc = 1.0 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 -0.1 0.20 0.17 1.75 1.40 1.39 1.53 -0.2 0.16 0.13 1.22 1.04 0.58 1.12 Analysis of Non-symmetrical Airfoils and their Configurations 210 Appendices Table B.16: ML configuration results for -0.4 ≤ a f ≤ 0.4 Single-sided xfc = 0.0 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 0.1 0.22 0.22 2.03 1.94 0.57 1.67 0.2 SD SD SD SD SD SD xfc = 0.5 af h Ct Cl NACA6302 S1020 NACA6302 S1020 NACA6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 0.1 0.20 0.23 1.65 1.83 1.35 3.13 0.2 0.22 SD 1.63 SD 2.68 SD 0.3 SD ND SD ND SD ND xfc = 1.0 af h Ct Cl NACA6302 S1020 NACA6302 S1020 N6302 S1020 0.0 0.21 0.20 1.78 1.63 1.72 1.93 -0.1 SD 0.21 SD 1.72 SD 3.57 -0.2 ND 0.19 ND 1.69 ND 4.38 -0.3 ND SD ND SD ND SD Analysis of Non-symmetrical Airfoils and their Configurations 211 Appendices C Instructions to Execute Codes C.1 UVLM User Instructions The UVLM program runs under windows XP. It can be installed by clicking on the Setup1.7.1.exe file§§§§§. The program can be run by clicking on the UVLMTest 1.7.1. A data file “xxxxxxx.dat” (for example HPlate.dat) is used to define the flapping motions. It is a text file and the user can modify it to define the wing size, curvature and flapping motions. A number of sample files can be found in the subdirectory of the program in “Test_Cases”. The GUI of the UVLM program is shown in Figure C.1. Figure C.1: GUI of the UVLM program By clicking on the “Run” button (circled), the program will start running. When the program has completed, click on “Display total force chart” (circled) to obtain the forces and torque §§§§§ Note that in some cases, the user might need to install Compaq Visual Fortran due to missing dll files. Analysis of Non-symmetrical Airfoils and their Configurations 212 Appendices generated, as shown in Figure C.2. Figure C.2: Total force chart of the UVLM program C.2 SCNSS User Instructions C.2.1 Compilation The source files of the SCNSS code with and without morphing both have the same names. However, the content of some of the files are different. They are global.F, grid.f90, flux_area.f90, bc.f90, bc_impl.f90, bc_semi.f90, set_matrix.f90, inter_step.f90, mom_disz.f90, poisson.f90, petsc_sub.F, cell_data.f90, fractional.f90 and ns2d_c.f90. The library files are tecio64.a and linux64.a and PETSc (Balay et al., 2003). PETSc is a library of linear solvers. The user has to download and compile the library before it can be used (http://wwwunix.mcs.anl.gov/petsc/petsc-as/index.html). The makefile also has to be slightly modified due to installation directory difference. The code is compiled using make –f makefile_scnss. The same procedure is used for IBCNSS codes as well. It is recommended that the user use the precompiled a.out which is statically linked in the respective directories. Analysis of Non-symmetrical Airfoils and their Configurations 213 Appendices C.2.2 Execution The a.out is executed on the atlas3 server, which is one of the fastest clusters in SVU. The command is: bsub -a mvapich –o log -q linux64 -m "multicore" mpirun.lsf ./a.out $1 $2 … $12 where log is the logging file and $1 to $12 are the input variables. The usual practice is to first run the code with $12 equal to a large negative value. After the first run, one can estimate the number of time steps required for approximately one period of flapping. Thereafter, the second run is carried out with $12 known. C.2.2.1 Without Morphing The filename for the grid used is gridgen.grd. The code is executed using a.out $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 $11 $12 where $1 to $12 refers to the input variables given in Table C.1. Input $1 Table C.1: Input variables for SCNSS without morphing (continue on next page) Variables represented = Fresh start = Continue from last stop $2 = Not used $3 h0 $4 f $5 -q0 $6 f $7 y $8 = Not used $9 crot Analysis of Non-symmetrical Airfoils and their Configurations 214 Appendices $10 CFL number $11 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams Time steps required, with generation of vorticity diagrams******, negative $12 value if no vorticity diagram required C.2.2.2 With Morphing In this case, the grid’s filename is still the same. However the input variables are different. The code is executed using a.out $1 $2 $3 $4 $5 $6 $7 $8 $9 $12 where $1 to $12 refers to the input variables given in Table C.2. Input $1 Table C.2: Input variables for SCNSS with morphing (continue on next page) Variables represented = Fresh start = Continue from last stopping time step $2 = Run with pure heaving configuration = Run with ME configuration = Run with ME(20o) configuration = Run with MT configuration = Run with ML configuration $3 xfc $4 af ****** †††††† The difference between $12 and $11 is slightly more than the time steps required to obtain one flapping cycle. This will ensure the vorticity diagrams generated cover one flapping cycle. †††††† Currently $4 to $7 are all equal to a f . It is meant for further revision of the program. Analysis of Non-symmetrical Airfoils and their Configurations 215 Appendices $5 af $6 af $7 af $8 = 90.0 (fixed in this study) $9 = (normal flexing) = (single-sided flexing) $10 CFL number $11 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams $12 Time steps required, with generation of vorticity diagrams, negative value if no vorticity diagram required C.2.3 Output The output files and their descriptions are given in Table C.3. The coef.txt is to be imported into Microsoft excel file SCNSS.xls. The user has to input the correct flapping configurations and starting time (obtainable from time.txt). One also has to ensure that the plot represents integer number of periods so that the efficiency can be calculated correctly. Output coef.txt Table C.3: Output files for SCNSS (continue on next page) Description Contains data about lift, drag and moment coefficient. To be imported into Microsoft excel file SCNSS.xls time.txt Starting time and other values time_m.txt Instantaneous time and time steps config.txt Flapping configurations Analysis of Non-symmetrical Airfoils and their Configurations 216 Appendices node_value.txt Binary file of the node velocity values uv_value.txt Binary file of the cell center velocity values p_value.txt Binary file of the cell center pressure values vel_f_mn_value.txt Binary file of the face center grid velocity values vel_f_value.txt Binary file of the face center velocity values xy_value.txt Binary file of the x/y coordinates values node_v.plt Tecplot output of the flapping airfoil at the instance the code stops node01-30.plt Tecplot output of the flapping airfoil from time step = $11 to $12 for Table C.1. Used for vorticity diagrams Column Table C.4: Description for time.txt Description Time elapsed Starting time Total time steps Average outflow velocity Number of grid point in x direction Number of grid point in y direction 7-14 Miscellaneous values, used to resume calculation Column Table C.5: Description for time_m.txt Description Time elapsed Current time step Average outflow velocity Analysis of Non-symmetrical Airfoils and their Configurations 217 Appendices C.2.3.1 Without Morphing Column Table C.6: Description for config.txt (without morphing) Description Not used h0 f -q0 f y Not used currently crot CFL number 10 Interval between time steps at writing each vorticity diagram file 11 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams 12 Time steps required, with generation of vorticity diagrams, negative value if no vorticity diagram required C.2.3.2 With Morphing Column Table C.7: Description for config.txt (with morphing) (continue on next page) Description Not used h0 f -q0 f xfc Analysis of Non-symmetrical Airfoils and their Configurations 218 Appendices af af af 10 af 11 yf 12 CFL number 13 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams 14 Time steps required, with generation of vorticity diagrams, negative value if no vorticity diagram required C.3 IBCNSS User Instructions C.3.1 Compilation The source files of the IBCNSS code for or airfoils in tandem both have the same names. However, the content of some of the files are different. The source files of the code are global.F, grid.f90, flux_area.f90, bc.f90, bc_impl.f90, bc_semi.f90, set_matrix.f90, inter_step.f90, mom_disz.f90, poisson.f90, airfoil.f90, hypre.f90, cell_data.f90, fractional.f90 and ns2d_c.f90. The library file is tecio64.a. The code is compiled using make –f makefile_ibcnss. Similar, the user is encouraged to use the a.out in the IBCNSS directory. C.3.2 Execution The a.out is executed on the atlas3 mcore parallel or atlas4 quad_parallel server since it is Analysis of Non-symmetrical Airfoils and their Configurations 219 Appendices meant to run on more than processor. The airfoils or body shapes are determined by body.txt and body2.txt for tandem arrangements. body.txt and body2.txt are ASCII files which contains the numbers of pts and the body coordinates. C.3.2.1 For Airfoil The code is executed using a.out $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 where $1 to $10 refers to the input variables given in Table C.8. Input Table C.8: Input variables for IBCNSS for airfoil Variables represented $1 = Fresh start = Continue from last stopping time step $2 Grid number, in x direction, in multiples of 60 $3 Grid number, in y direction, in multiples of 36 $4 h0 $5 f $6 -q0 $7 f $8 CFL number $9 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams $10 Time steps required, with generation of vorticity diagrams, negative value if no vorticity diagram required C.3.2.2 For Airfoils in Tandem The code is executed using a.out $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 where $1 to $10 refers to the input variables given in Table C.9. Analysis of Non-symmetrical Airfoils and their Configurations 220 Appendices Input $1 Table C.9: Input variables for IBCNSS for airfoils in tandem Variables represented = Fresh start = Continue from last stopping time step $2 = Used for grids with d12 ≤ 2.5 = Used for grids with d12 > 2.5 $3 Grid number, in x direction, in multiples of 110 if $2 = 4, 132 if $2 = $4 Grid number, in y direction, in multiples of 70 $5 = Run with ME configuration = Run with MT configuration = Run with ML configuration $6 d12 $7 f12 $8 CFL number $9 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams $10 Time steps required, with generation of vorticity diagrams, negative value if no vorticity diagram required C.3.3 Output The output files and their descriptions are given in Table C.10. The coef3.txt is to be imported into Microsoft excel file IBCNSS.xls. The user has to input the correct flapping configurations and starting time. One also has to ensure that the plot represents integer number of periods so that the efficiency can be calculated correctly. Analysis of Non-symmetrical Airfoils and their Configurations 221 Appendices Output Table C.10: Output files for IBCNSS Description coef3.txt Contains data about lift, drag and moment coefficient. To be imported into Microsoft excel file IBCNSS.xls time.txt Starting time and other values time2.txt Starting time and other values time_m.txt Instantaneous time and time steps config.txt Flapping configurations uv_value.txt Binary file of the cell center velocity values p_value.txt Binary file of the cell center pressure values body_pts.txt Binary file of the fore airfoil instantaneous coordinates body_pts2.txt Binary file of the aft airfoil instantaneous coordinates node01-20.plt Tecplot output of the flapping airfoil from time step = $9 to $10 for Table C.8. Used for vorticity diagrams Column Table C.11: Description for time.txt Description Time elapsed Starting time Total time steps Average outflow velocity Number of grid point in x direction Number of grid point in y direction 7-14 Miscellaneous values, used to resume calculation Column Table C.12: Description for time2.txt Description 1-7 Miscellaneous values, used to resume calculation Analysis of Non-symmetrical Airfoils and their Configurations 222 Appendices Column Table C.13: Description for time_m.txt Description Time elapsed Current time step Average outflow velocity Column Table C.14: Description for config.txt Description Not used h0 f -q0 f y crot d12 f12 10 CFL number 11 Interval between time steps at writing each vorticity diagram file 12 Time steps required if no generation of vorticity diagrams required or starting time step for generation of vorticity diagrams 13 Time steps required, with generation of vorticity diagrams, negative value if no vorticity diagram required Analysis of Non-symmetrical Airfoils and their Configurations 223 [...]... amplitudes are tested Moreover, the effect of center of flexure, leading/trailing edge flexing and the use of non- symmetrical airfoil are also investigated Hence, the parameter Analysis of Non- symmetrical Airfoils and their Configurations 3 1 Introduction space is now much larger The last phase of the research investigates the effect of the arrangement of the airfoils in tandem Through simulations, one hopes... W Angular velocity of the body fixed frame in the inertial frame G Circulation Dq Induced velocity Analysis of Non- symmetrical Airfoils and their Configurations xvi 1 Introduction 1 Introduction The objective of this research is to enhance the understanding of flapping- wing mode of flying The ultimate aim is really to improve the performance of the efficiency, thrust and lift of flapping wing aircraft... 158 Figure C.1: GUI of the UVLM program 212 Figure C.2: Total force chart of the UVLM program 213 Analysis of Non- symmetrical Airfoils and their Configurations xi List of Tables Table 3.1: Comparison between Tuncer and Kaya’s and current results 46 Table 3.2: Flapping parameters for comparison against experimental results, taken from Figure 6 to 9 of Anderson et al (1998)... complex systems of unsteady Analysis of Non- symmetrical Airfoils and their Configurations 1 1 Introduction vortices Analysis of these results is therefore not easy Moreover, besides investigating the different flapping configuration, research has also branched into other areas to further improve the performance of flapping wing configurations These include 1 Active chordwise flexing (Miao and Ho 2006)... difference between rowing and heaving, in degrees Analysis of Non- symmetrical Airfoils and their Configurations xv yf Phase angle between plunging and flexing of airfoil, in degrees f Phase difference between pitching and heaving, in degrees f12 Phase difference between heaving position of first and second airfoils, in degrees h Propulsive efficiency ho Overall propulsive efficiency for tandem configuration... 3D flapping motion of a tethered sphingid moth and compared their results with the quasi-steady and the experimental Analysis of Non- symmetrical Airfoils and their Configurations 10 2 Literature Review ones It was found that their result was much more accurate than the quasi-steady one and closer to the experimental values It was mentioned that the unsteady panel method is valid for flow with Re of. .. about the effect of reduced frequency on thrust Wu and Sun (2005) Analysis of Non- symmetrical Airfoils and their Configurations 11 2 Literature Review studied the effect of wake on the aerodynamics forces It was found that at the start of the halfstroke, the wake might either increase or decrease the lift and drag It depended on the kinematics of the wing at stroke reversal For the rest of the half-stroke,... interaction study to investigate the effect of chordwise and spanwise flexing on a flapping foil The foil was simulated to be immersed in two different types of fluids of high and low density It was found that in low density fluid, the chordwise flexibility reduced both the thrust and efficiency, while the spanwise flexibility Analysis of Non- symmetrical Airfoils and their Configurations 14 2 Literature Review... on the other hand, investigated on the effects of different angle of attack profiles Both the sawtooth and cosine profiles showed improvement in thrust coefficient or efficiency over the standard sinusoidal profile This showed that besides the selection of certain parameters as mentioned earlier, different types of flapping profiles such as sawtooth also influenced the performance of the airfoils Schouveiler... investigate flapping configurations which not only give high efficiency and thrust, but also high lift through the use of non- symmetrical airfoils This method of generating lift is much more advantageous than changing the stroke angle to produce thrust/lift through force vectoring, assuming the same flapping parameters are used In this study phase, a total of four other non- symmetrical airfoils are used The airfoils . Efficiency of Different Airfoils 94 5.3 Significance and Effect of Variables on Thrust 95 5.3.1 Significance of k and q 0 and their Interaction 97 5.3.2 Significance of k and f and their Interaction. Analysis of Non-symmetrical Airfoils and their Configurations vi 4.3 Tandem Airfoils 76 5 Results and Discussions from the DOE 84 5.1 The Box-Behnken (BB) Test 86 5.2 Significance and. C.3.2.2 For 2 Airfoils in Tandem 220 C.3.3 Output 221 Analysis of Non-symmetrical Airfoils and their Configurations ix List of Figures Figure 2.1: A representation of the airfoil