Aerodynamics and Design for Ultra-Low Reynolds Number Flight

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Chapter 3 laminar flat plate drag differs from the linear extrapolated zero thickness drag by only 5.3% at Re=6000 and 9.5% at Re=2000. This result is not surprising considering that the zero thickness reference is a flat plate. This supports the assumption of a linear relationship between thickness and zero lift drag for these airfoils. The drag penalty associated with increasing thickness grows as the Reynolds number is decreased, but the rates of drag increase relative to the zero thickness drag are similar. Although the magnitudes of the drag coefficients increase dramatically, they are in line with simple laminar plate results. The variations with thickness also exhibit an order of magnitude increase, but the trends are consistent with those at higher Reynolds numbers. 3.3.2 Effect of Thickness on Lift Within the quasi-linear range, the inviscid lift curve slope of an airfoil benefits from increased thickness, with thicker sections obtaining as much as a 10% increase in lift curve slope over the thin airfoil value of 2π per radian. At more conventional Reynolds numbers, viscous effects then degrade the lift curve slope, with the net result of lift curve slopes 5% to 10% below the inviscid thin airfoil value. The increased thickness of the upper surface boundary layer relative to the lower surface boundary layer at a positive angle of attack effectively reduces the camber of the airfoil or equivalently, the angle of attack. The inviscid gains and viscous losses typically cancel, resulting in lift curve slopes close to 2π per radian across a range of section thickness. This is not the case for Reynolds numbers below 10,000. In this range of operation, the viscous boundary layer growth dominates and increasing thickness results in a significant decrease in lift curve slope in the quasi-linear range. The results in Figure 3.8 show as much as a 35% reduction in lift curve slope for the 8% thick section from the idealized value of 2π. The 2% thick sections come closest to the inviscid thin airfoil value, showing a 15% reduction. The effect of reducing the Reynolds number from 6000 to 2000 is a further reduction in lift curve slope. In Figure 3.9, the 2% section shows the greatest effect with a 5.2% reduction relative to the 32
Re=6000 result due to Reynolds number effects. The effect of Reynolds number variations decreases with increasing thickness. Cl 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 NACA 0002 NACA 0004 NACA 0006 NACA 0008 -0.20 -2 -1 0 1 2 3 α (deg.) 4 5 6 7 8 FIGURE 3.8 Lift curves for uncambered NACA 4-digit sections, Re=6000. Chapter 3 33
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AERODYNAMICS AND DESIGN FOR ULTRA-L
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I certify that I have read this dis
Page 7 and 8: Abstract Growing interest in micro-
Page 9 and 10: Nomenclature A Rotor disk area B Nu
Page 11 and 12: τ Shear stress ω, Ω Rotor angul
Page 13 and 14: Contents Acknowledgments...........
Page 15 and 16: Chapter 5..........................
Page 17 and 18: List of Tables 3.1 Effect of Airfoi
Page 19 and 20: List of Figures 2.1 Comparison of I
Page 21 and 22: 5.6 Small test fixture in the large
Page 23 and 24: 6.38 Predicted spanwise thrust dist
Page 25 and 26: Chapter 1 Introduction 1.1 Looking
Page 27 and 28: Chapter 1 wind-tunnel facilities an
Page 29 and 30: 1.3 Summary of Contributions Chapte
Page 31 and 32: Chapter 2 Two-Dimensional Computati
Page 33 and 34: Chapter 2 The artificial compressib
Page 35 and 36: Chapter 2 The control volume method
Page 37 and 38: 2.2.4 Comparison with Experiment Th
Page 39 and 40: Chapter 2 to diverge if significant
Page 41 and 42: 2.4 Flow Field Assumptions The INS2
Page 43 and 44: Chapter 2 The results of these time
Page 45 and 46: Chapter 3 Analysis and Design of Ai
Page 47 and 48: Chapter 3 adverse gradient in the p
Page 49 and 50: The canonical pressure coefficient
Page 51 and 52: Chapter 3 reaches α=5.0 and a lift
Page 53 and 54: ��
Page 55: Cl 0.50 0.40 0.30 0.20 0.10 0.00 -0
Page 59 and 60: FIGURE 3.10 NACA 0002 and NACA 0008
Page 61 and 62: C l C l 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 63 and 64: Chapter 3 The maximum L/D for all n
Page 65 and 66: 3.5 Effect of Leading Edge Shape an
Page 67 and 68: C l 0.60 0.50 0.40 0.30 0.20 0.10 0
Page 69 and 70: Chapter 3 prominent droop near the
Page 71 and 72: Chapter 3 Small improvements in lif
Page 73 and 74: Chapter 4 Hybrid Method for Rotor D
Page 75 and 76: Q/r D L Chapter 4 FIGURE 4.1 Typica
Page 77 and 78: 4.2.3 Elimination of Trigonometric
Page 79 and 80: Chapter 4 applications of interest
Page 81 and 82: 4.2.7 The Distinction Between the A
Page 83 and 84: Chapter 4 Two input parameters dete
Page 85 and 86: Chapter 4 simplification, particula
Page 87 and 88: 4.3.3 Conservation of Angular Momen
Page 89 and 90: Chapter 4 moves downstream is less
Page 91 and 92: Chapter 4 of the pitch angle at the
Page 93 and 94: FIGURE 4.6 Converged wake streamlin
Page 95 and 96: Chapter 4 idealized models. The met
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Page 99 and 100: Chapter 5 Overview of Experimental
Page 101 and 102: 1. Substrate Preparation 2. Polymer
Page 103 and 104: Chapter 5 surface finish and aids i
Page 105 and 106: FIGURE 5.5 Small test fixture in th
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Chapter 5 The direct influence of t
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FIGURE 5.9 Large test fixture in th
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5.4.1 Motor Power Supply All tests
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Chapter 5 or roughly (+/-) 25% of t
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Chapter 5 newly forming blade tip v
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FIGURE 5.14 Near-body grid geometry
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Chapter 6 Design Examples and Compa
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Chapter 6 capable of speeds up to 1
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FIGURE 6.2 The Astro Flight Firefly
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t/c 0.5 0.4 0.3 0.2 0.1 0.0 FIGURE
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c/R 0.0 0.2 0.4 0.6 0.8 1.0 r/R Cha
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FIGURE 6.10 Carbon-fiber two-blade
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Power Req. (W) 3.5 3.0 2.5 2.0 1.5
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Chapter 6 TABLE 6.1 Comparison of p
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Electro-mechanical Efficiency 0.20
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these variations and possible solut
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upper and lower surface corners of
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Chapter 6 The Sample-1 rotor exhibi
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The static deformations described i
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Q( r) This problem is solved as a l
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Structural Analyses Chapter 6 The m
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Aero-Structural Twist (deg.) 0.00 -
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Thrust (g) 5 4 3 2 1 0 10000 20000
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Thrust (g) 4.5 4.0 3.5 3.0 2.5 2.0
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Torque per unit span (g cm) 0.75 0.
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Thrust per unit span (g) 3.5 3.0 2.
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V total / ωr 1.0 0.9 0.8 Classical
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Torque per unit span (g cm) 0.75 0.
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Chapter 6 Recall that this rotor wa
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Incidence (deg.) 30 25 20 15 10 5 0
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Thrust (g) 9 8 7 6 5 4 3 2 38000 40
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x/R 0.4 0.2 0.0 -0.2 Chapter 6 FIGU
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Cl 0.64 0.62 0.60 0.58 0.0 0.2 0.4
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C f 0.125 0.100 0.075 0.050 0.025 0
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Chapter 7 Micro-Rotorcraft Prototyp
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This vehicle does not currently inc
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FIGURE 7.2 The 65g prototype electr
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7.4 The 150g Prototype Chapter 7 Th
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At hover, with a mass of 153g, the
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Using the 150g prototype as an exam
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Chapter 7 Basic rotor theory has be
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most of the blade. These values are
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Chapter 7 For large scale helicopte
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Chapter 8 Conclusions and Recommend
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the significant performance gains a
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Chapter 8 are also opportunities to
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Chapter 8 formation flight, and man
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References [1] Spedding, G.R., Liss
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[28] Lamb, Sir Horace, Hydrodynamic
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Aerodynamics and Design for Ultra-Low Reynolds Number Flight

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