Lift Distributions on Low Aspect Ratio Wings at Low Reynolds Numbers by Sagar Sanjeev Sathaye A Thesis Submitted to the Faculty of WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Master of Science in Mechanical Engineering by _____________________________ May 2004 Approved: ________________________________________________________ Professor David J Olinger, Thesis Advisor ________________________________________________________ Professor Hamid Johari, Committee Member ________________________________________________________ Professor William Durgin, Committee Member ________________________________________________________ Professor John Sullivan, Graduate Committee Representative Abstract The aerodynamic performance of low aspect ratio wings at low Reynolds numbers applicable to micro air vehicle design was studied in this thesis. There is an overall lack of data for this low Reynolds number range, particularly concerning details of local flow behavior along the span. Experiments were conducted to measure the local pressure distributions on a wing at various spanwise locations in a Reynolds number range 3×104 < Re < 9×104. The model wing consisted of numerous wing sections and had a rectangular planform with NACA0012 airfoil shape with aspect ratio of one. One wing section, with pressure ports at various chordwise locations, was placed at different spanwise locations on a wing to effectively obtain the local pressure information. Integration of the pressure distributions yielded the local lift coefficients. Comparison of the local lift distributions to optimal elliptic lift distribution was conducted. This comparison showed a sharply peaked lift distribution near the wing tip resulting in a drastic deviation from the equivalent elliptic lift distributions predicted by the finite wing theory. The local lift distributions were further analyzed to determine the total lift coefficients vs angle of attack curves, span efficiency factors and the induced drag coefficients. Measured span efficiency factors, which were lower than predictions of the elliptic wing theory, can be understood by studying deviations of measured lift from the elliptic lift distribution. We conclude that elliptic wing theory is not sufficient to predict these aerodynamic performance parameters. Overall, these local measurements provided a better understanding of the low Reynolds number aerodynamics of the low aspect ratio wings. 2 Acknowledgements I would like to thank my advisor Professor Olinger for all the guidance, kindness and belonging he has given me over the last two years. He had envisaged the idea of making a wing in sections and using it for pressure measurements in such an innovative and simple way. I am glad to be a part of his imagination becoming a reality. I would also like to thank Professor Bill Weir here for his continued support and encouragement and his fabulous participation in making the wing. Right from day one Professor Weir’s interest and involvement in the manufacturing of the wing was incredible. I sincerely thank Professor Olinger and Professor Weir for successful completion of my research. I would also like to thank Professor Johari for all the discussions and timely advice over the course of this research. I also thank Professor Durgin and Professor Sullivan for serving as my defense committee members. I thank Abhijeet, Siju, Elham, Gana and Ian for their support and affection without whom the last two years wouldn’t have been so enjoying and fulfilling. I thank Ryan and Shang for sharing all the hilarious moments, which made my stay at WPI a lot comfortable. I thank Jim Johnston whom I bothered till the last day of my research, without ever being denied a helping hand. Last and definitely not the least I am really grateful to my parents who have been my inspiration and strength over all these years and Mugdha and Sarang for being so supportive and encouraging, and being there for me when I needed. 3 To Aai – Baba 4 Table of Contents: List of Figures……………………...……………………………………………………iii List of Tables……….………………...…………………………………………….…….v Nomenclature…..……………………..………………………………………….……...vi 1. Introduction……..……………………………………………………………………..1 1.1 Background………………………………………………………………..1 1.2 Previous work……………………………………………………………..3 1.3 Thesis Objectives………………………………………………………….8 2. Methods…………..…………………………………………………………………...10 2.1 Theory behind the analysis……………………………………………….10 2.2 Experimental set-up………………………………………………………16 2.2.1 Wind tunnel………………………………………………………...16 2.2.2 Pressure Transducer………………………………………………..17 2.2.3 Wing Model………………………………………………………..17 2.2.4 Sting arm/ Angle of attack control mechanism…………………….23 2.2.5 Lift measurement set-up using force balance...………………..…..23 2.3 Tygon tube out-gassing…………………………………………………..24 2.4 Velocity Distribution across wind tunnel………………………………..25 2.5 Experimental Procedure………………………………………………….27 2.6 Error analysis…………………………………………………………….28 3. Results………...………………………………………………………………………29 3.1 Local Pressure measurements……………………………………………29 3.2 Spanwise Lift Distributions……………………………………………...47 i 3.3 Fourier Coefficients………………………………………………………59 3.4 Span efficiency factor…………………………………………………….63 3.5 Lift coefficient curve vs angle of attack………………………………….68 3.6 Induced drag coefficients…………………………………………………73 4. Conclusions………...…………………………………………………………………77 5. Suggestions for future work…………………………………………………………81 References…..…………………………………………………………………………..84 Appendix A Pressure distributions……...……………………………………………....86 Appendix B MATLAB Codes……………………………...………………………….109 Appendix C Fourier Coefficients……………………………..………………………..114 Appendix D Sample Error calculation……………………………..…………………..117 i i List of Figures: Figure 2.1 Exploded view of the low aspect ratio test wing (NACA0012)……………...19 Figure 2.2 Numbering scheme for spanwise locations for the pressure section…………19 Figure 2.3 Chord wise locations of ports on the pressure section……………………….20 Figure 2.4 Numbering scheme for the ports……………………………………………..21 Figure 2.5 Experimental set-up (schematic)……………………………………………..22 Figure 2.6 Velocity distributions across wind tunnel……………………………………26 Figure 3.1 C vs x/c plots for Re = 35966, α = 15o……………………………………...34 p Figure 3.2 C vs x/c plots for Re = 35966, α = 6o……………………………………….35 p Figure 3.3 Flow visualization from Torres and Mueller [17] ……………………...……36 Figure 3.4 C vs x/c plots for Re = 30218, α = 15o…...…………………………………37 p Figure 3.5 C vs x/c plots for Re = 30218, α = 6o…....………………………………….38 p Figure 3.6 C vs x/c plots for Re = 84122, α = 15o……..……………………………….39 p Figure 3.7 C vs x/c plots for Re = 84122, α = 6o……....……………………………….40 p Figure 3.8 C vs x/c plots for Re = 49345, α = 15o……..……………………………….41 p Figure 3.9 C vs x/c plots for Re = 49345, α = 6o…...….……………………………….41 p Figure 3.10 Effect of variation of α on Cp distribution for Re = 43615………………...43 Figure 3.11 Effect of variation of Re variation on Cp distribution for α = 6o…………..44 Figure 3.12 Effect of variation of Re variation on Cp distribution for α = 15o…………45 Figure 3.13 Cp vs x/c plots at α = 0o for Pos 1…………………………………………..46 Figure 3.14 Local lift coefficient distribution at α = 15o………………………………..51 Figure 3.15 Local lift coefficient distribution at α = 6o…………………………………52 i i i Figure 3.16 Approximate Wing tip vortex core location ………………………………..53 Figure 3.17 Local lift distribution at α = 15o……………..……………………………...54 Figure 3.18 Local lift distribution at α = 6o…………………..………………………….55 Figure 3.19 Normalized lift distributions for Rectangular wings with varying AR……..56 Figure 3.20 Local lift coefficient distribution for various angles of attacks……………..57 Figure 3.21 Local lift coefficient distribution for Re = 30218 at α = 15o……………….58 Figure 3.22 Span efficiency factor vs No. of coefficients used for analysis…………….62 Figure 3.23 Variation of span efficiency factor with Reynolds number…………………65 Figure 3.24 Variation of span efficiency factor with angle of attack……………………66 Figure 3.25 Spanwise lift distribution for Re = 43615………..…………………………67 Figure 3.26 Lift vs angle of attack curve at Re = 43615………………………………...71 Figure 3.27 Lift vs angle of attack curve at Re = 84122………………………………...72 Figure 3.28 Induced drag coefficients with varying α…………………………………..75 Figure 3.29 Induced drag coefficients with varying Re………………………………….76 Figure 5.1 Adjusted chord distribution…………………………………………………..83 i v List of tables: Table 2.1: Spanwise distances from the wing tip for pressure section…………………..20 Table 2.2: Chordwise locations of ports as percent chord (x/c)…………………………21 Table 2.3: Experimental matrix………………………………………………………….27 v Nomenclature: AR = aspect ratio of the wing. α = angle of attack A = Axial Force b = span of the wing. c = chord length of the wing. c = local axial coefficient. a c =local drag coefficient d C = total drag coefficient. D C = parasitic Drag coefficient. Do C = induced drag coefficient Di c = shear stress coefficient. f c = local lift coefficient l C = total lift coefficient. L c = local normal coefficient. n C = pressure coefficient. p D = Drag. e = span efficiency factor. Г = circulation l = lower surface quantity L = lift. L’ = lift per unit span (local lift) µ = Absolute Viscosity of air (1.79 e-5) v i