A Discrete Vortex Method Application to Low Reynolds Number Aerodynamic

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A DISCRETE VORTEX METHOD APPLICATION TO

LOW REYNOLDS NUMBER AERODYNAMIC FLOWS

Thesis

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Master of Science in Aerospace Engineering

Patrick R. Hammer

University of Dayton

Dayton, OH

August 2011

i A DISCRETE VORTEX METHOD APPLICATION TO LOW REYNOLDS NUMBER

AERODYNAMIC FLOWS

Name: Hammer, Patrick Richard

APPROVED BY:

______Aaron Altman, PhD Greg Reich, PhD Advisory Committee Chairperson Committee Member Associate Professor Adaptive Structures Team Lead Department of Mechanical and Aerospace Eng. Air Vehicles Directorate

______Frank Eastep, PhD Advisory Committee Chairperson Professor Emeritus Department of Mechanical and Aerospace Eng.

______John G. Weber, PhD Tony E. Saliba, PhD Associate Dean Dean, School of Engineering School of Engineering & Wilke Distinguished Professor

ii ABSTRACT

A DISCRETE VORTEX METHOD APPLICATION TO LOW REYNOLDS

NUMBER AERODYNAMIC FLOWS

Name: Hammer, Patrick R. University of Dayton

Advisor: Dr. Aaron Altman

Although experiments and CFD are very powerful tools in analyzing a niche of fluid dynamics problems relevant to developing Micro Aerial Vehicles (MAVs), reduced order methods have shown to be very capable in helping researchers achieve a basic understanding of flow physics with application to highly iterative design processes due to the less computationally expensive nature of the low order models. The current study used one low order method, the Discrete Vortex Method, to model the aerodynamic flow fields and forces around a thin airfoil undergoing a variety of flows, as well as parametric studies to determine the important factors that had to be adjusted to make the results more representative of the physical phenomenon being modeled. Initial investigations validated the code’s use in steady flow and low amplitude unsteady flow cases by comparing it with circulation distributions of various airfoil shapes, the Wagner function, and Theodorsen’s function. The results showed a strong dependency on bound vortex number and time step size. The code was then used to capture the flow behavior around the airfoil for various AIAA Fluid Dynamics Technical Committee Low Reynolds

iii

Number Working Group (FDTC-LRWG) canonical cases. Implementing the Uhlman method in the Discrete Vortex Method allowed for the calculation of the pressure at the airfoil surface and in the flow field during high angle attack maneuvers. This method proved very capable in calculating the pressures, forces, and force coefficients around the airfoil post-flow separation in the canonical cases where other methods (such as the

Unsteady Bernoulli Method) fall short. The code was also tuned with respect to the results with respect to vortex size, leading edge separation strength factor, desingularization function, wake radius size factor, and in the Uhlman method itself to yield an optimal comparison with experimental and CFD results. The study found a bound vortex number of 30, a leading edge separation strength factor of 1.0, the planetary desingularization function, a wake radius size factor of 1.0, and using just the volume integral term on the RHS of the Uhlman method gave the best results for the geometry analyzed. An investigation then determined the dependency of reduced frequency on the lift and drag coefficients for the canonical cases. Finally, the code was used to model a

“true perch” by implementing a curve fit function which caused the horizontal free stream velocity to decrease to zero. In this context, the forces were of more interest than the force coefficients since the coefficients experienced anomalous behavior as the free stream velocity approached zero. It was also interesting to find that the code modeled behavior very similar to shear layer instabilities in the LE and TE shear layers, caused by a rippling effect as the bound circulation changed in strength and sign as the LEV and

TEV interacting with it. Recommendations were then made to apply the code to airfoils with either fixed or variable camber since camber acts as a high lift device and could prove very beneficial in the design and development of MAVs

iv ACKNOWLEDGEMENTS

I would first like to thank Dr. Aaron Altman for his tremendous leadership as both an academic, an advisor, and in the aerospace field. Without him, I would not have been brought to this very interesting project with respect to MAV perching, nor would I have developed my interest in aerospace engineering to a passion. I would also like to acknowledge his wife, Servane, his son, Samuel, and his twin daughters, Eloise and

Melodie.

I would next like to thank my committee members, Dr. Greg Reich and Dr. Frank

Eastep. I would especially like to thank Dr. Eastep for his introducing me to the Discrete

Vortex Method, which was subject of this thesis.

I would next like to thank Dr. Darrel Robertson for his tireless help throughout all of my coding issues, as well as teaching me how to implement various aerodynamic concepts into MATLAB. Without his help, this thesis would not have been completed.

I would next like to thank Dr. James Joo for his help as UDRI liaison before joining AFRL as a civilian employee.

I would like to thank Dr. Michael Ol for giving me data for which I could compare my low order code with.

I would also like to thank my fellow students, Ethan Harper, Ben Hager, Frank

Semelmayer, Matt Geyman, John Puttmann, and Danielle Christenson.

v I would like to thank my family for their financial and emotional support during all of my academic years.

I would finally like to thank Michigan State University for preparing my intellect for the coursework and research that I completed at the University of Dayton. Go

Spartans!

vi TABLE OF CONTENTS

ABSTRACT...... iii

ACKNOWLEDGEMENTS...... v

LIST OF FIGURES...... ix

LIST OF TABLES...... xx

LIST OF SYMBOLS/ABBREVIATIONS...... xxi

CHAPTER 1 - INTRODUCTION...... 1

1.1 Background...... 1

1.2 Literature Review...... 5

CHAPTER 2 – THIN AIRFOIL THEORY AND THE DISCRETE VORTEX

METHOD...... 21

2.1 Potential Flow Theory and Thin Airfoil Theory...... 21

2.2 Discrete Vortex Method ...... 33

vii CHAPTER 3 – STEADY FLOW AND LOW ANGLE OF ATTACK UNSTEADY

AERODYNAMIC VALIDATION...... 47

3.1 Steady Flow Validation...... 47

3.2 Classical Unsteady Aerodynamics...... 51

3.3 Unsteady Flow Validation...... 59

CHAPTER 4 – HIGH ANGLE OF ATTACK CANONICAL

CASES...... 67

4.1 High Angle of Attack Flow Field Validation...... 67

4.2 The Uhlman Method...... 87

4.3 Pressure, Force, and Force Coefficient Calculations Using the Uhlman

Method...... 92

4.4 Reduced Frequency Dependency...... 120

4.5 Application of DVM Code and Uhlman Method to Perching Maneuver...... 127

4.6 Vortex/Shear Layer Instabilities...... 141

CHAPTER 5 – CONCLUSIONS AND RECOMMENDATIONS...... 148

REFERENCES...... 152

APPENDICES...... 156

A-1 Obstacles Encountered...... 156

viii LIST OF FIGURES

Figure 1-1: Vorticity plots for 40o ramp-hold-ramp case with reduced frequency k of 0.7 shows a good comparison between Ol (left), Lian (middle), and Eldredge (right)1. All three methods show a well defined trailing edge vortex during the pitch up while the leading edge vortex grows. As the plate pitches down, a counter-rotating vortex is shed from the trailing edge while the leading edge vortex continues to grow...... 6

Figure 1-3: Vorticity plots for 45o ramp-hold case with shows a good comparison between Ol (first and second columns), Garmann and Visbal (third-fifth columns)2. The experiments and CFD both show Kelvin-Helmholtz instabilities shed from the trailing edge as the airfoil begins the pitching maneuver. All five methods show a leading edge vortex that gets shed due to its interaction with the shear layer. As the trailing edge vortex gets shed, the leading edge vortex reattaches until shear layer interaction force it to de-attach...... 8

Figure 1-4. Lift and drag coefficient comparison between an SD7003 and a flat plate for various Reynolds numbers and maximum angle of attack2. The results showed a good comparison, except in the lift coefficient post stall, where there seemed to be a phase difference between the vortex shedding of the airfoil and flat plate...... 9

Figure 1-5: Various trajectories for the perching maneuver presented by Robertson4...... 11

Figure 1-6: Thrust ratio versus distance from the ground comparisons show that the QVLM and LVM both do a good job approximating the experimental results12...... 15

ix Figure 1-7: Comparison of four methods on the diffusing vortex sheet: random walk with a seed of 13(a), random walk with a seed of 19 (b), velocity diffusion (c), and vorticity redistribution (d)16...... 17

Figure 1-8: Aircraft model employed by Wickenheiser and Garcia that has the ability to morph for the purpose of MAV perching17...... 19

Figure 2-1. Thin airfoil approximation of infinitesimal vortex sheet placed on camber- line18...... 24

Figure 2-2. Streamlined velocity vectors are tangent to the airfoil surface...... 25

Figure 2-3. Diagram of the circulation around an airfoil when airfoil is at rest and at the beginning of forward motion18...... 30

Figure 2-4. Flow remains attached at low angles of attack; separates at high angles of attack when the fluid is forced to wrap around the leading edge21...... 32

Figure 2-5. Airfoil shape plot with the finite vortex and collocation points shown for n = 721...... 34

Figure 2-6. Airfoil shape plot which incorporates vortices shed from trailing edge21....37

Figure 2-7. Airfoil shape plot which incorporates vortices shed from leading edge21....41

Figure 2-8. Flow chart of DVM code...... 43

Figure 2-9. Various desingularization reduction factor plots show how the vortex influenced is reduced as vortices get close to each other22...... 45

Figure 3-2: Shape and circulation curves for the three test runs on a 10% cambered airfoil at 0o angle of attack to display the strong approximation when compared to the results

x using the Söhngen Inversion Method and Glauert Method. The coordinate system was shifted to the mid-chord since the Söhngen Inversion Method results were calculated with the mid-chord at the coordinate system origin...... 50

Figure 3-4. The Wagner Function plot versus s23...... 53

Figure 3-5. A comparison between the Wagner Function and the Lumped Vortex Method shows a very poor approximation at low convective times21...... 54

Figure 3-6. The wake shape generated from an impulsively started airfoil21...... 54

Figure 3-7. The two Wagner function approximations compared to the tabular data versus the convective time shows that the second approximation (which incorporates exponential functions) provides a better approximation...... 56

Figure 3-8. Theodorsen’s function for a purely plunging airfoil at k = 1.0, 0.5, and 0.1...... 58

Figure 3-9. Theodorsen’s Function for a purely pitching airfoil at k = 1.0, 0.5, and 0.1...... 58

Figure 3-10: The non-dimensionalized lift curves with Δt of 0.01 seconds for 1, 10, and 100 discrete vortices compared to the two Wagner approximations shows a very strong agreement between the two theoretical approximations and the result when n = 100...... 60

Figure 3-11: The non-dimensionalized lift curves with n = 100 for a Δt of 1, 0.1, and 0.01 compared to the two Wagner approximations shows a very strong agreement between the two theoretical approximations and the result when Δt = 0.1 and 0.01s...... 61

Figure 3-12: Transient lift curves compared to Theodorsen’s function and wake shapes for pure plunging at k = 1.0, 0.5, and 0.1 show very good comparison between the DVM and Theodorsen’s function...... 63

Figure 3-13: Re-scaled lift curves compared to Theodorsen’s function for pure plunging at k = 0.1 to show that the airfoil did experience transient lift at the low reduced frequency...... 64

Figure 4-1: Experimental results performed by Granlund, et. al.,2 compared to the numerical results of the leading edge and trailing edge vortex wake shapes for an airfoil impulsively started with α = 45o showed a good comparison between the experimental and the results obtained by the discrete vortex code...... 69

Figure 4-2: Angle of attack versus the physical time for the 0o-45o ramp-hold case presented by FDTC LRNDG2...... 70

Figure 4-3: Experimental (left), low order model (middle), and CFD (right) vorticity and flowfield results for the 45o ramp-hold from 0.6o to 45o show a good comparison2...... 75

Figure 4-4: Angle of attack versus the physical time for the 0o-40o-0o ramp-hold case presented1...... 76

Figure 4-6: Comparison of flow fields for three angles of attack by varying the number of bound vortices. The results show a more well-defined flow field as n was increased....79

xii

Figure 4-7: Comparison of flow fields for three angles of attack by varying the LE strength factor. The results show a more well-defined flow field as Ψ was decreased since it resulted in a smaller time step...... 81

Figure 4-8. Comparison of flow fields for three angles of attack by varying the flow separation angle of attack...... 83

Figure 4-9. Comparison of flow fields for three angles of attack by varying the shed vortex radius. The results show that varying the shed vortex radius did not have a significant impact on the vortex interaction...... 84

Figure 4-10: Comparison of flow fields for an angle of attack of 30o to display the effect of the six different desingularization functions shows that the Low Order Algebraic, Gaussian, and Super Gaussian function did not experience a significant amount of LEV instability that was experienced by the other functions...... 86

Figure 4-11: CL vs. convective time using the UBM shows good lift coefficient prediction until leading edge separation occurred...... 87

Figure 4-12: Points (blue) were created on the upper and lower surface of the airfoil so that the Uhlman method could be used with the DVM...... 89

Figure 4-13: CL, CD vs. convective time plot for the ramp hold case using the Uhlman method. The results show the Uhlman method gave an expected approximation throughout the entire convective time range. Although there are multiple peaks where von Karman shedding occurred, it can be expected that this was due to shear layer vortices causing the dips between near peaks...... 92

Figure 4-14: CL vs. angle of attack plot for the 0o-45o ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers and CFD data for a 0.2 span SD7003. Results show a good comparison, though the max lift coefficient using the DVM is higher than the results for the SD7003 at a Re of 50K2...... 95

Figure 4-15: CD vs. angle of attack plot for the 0o-45o ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers. Results show a good comparison, though the peaks and troughs during the von Karman shedding do differ between using the DVM is higher than the experimental results for the SD70032...... 96

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Figure 4-16: Comparison of lift coefficient versus convective time between DVM and CFD, analytical, and experimental models4. The results show a good overall comparison, though the occurrence in the DVM’s von Karman shedding was a result of the increased slope of the lift coefficient offsetting the stall lift coefficient with respect to convective time and shifting the occurrence of von Karman shedding...... 98

Figure 4-17: Comparison of lift coefficient versus convective time between DVM and time delay analytical4. The results show a good overall comparison, with a closer approximation to Analytical #2 than #1...... 99

Figure 4-18: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 18.75o...... 100

Figure 4-19: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 26.18o...... 100

Figure 4-20: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 33.7o...... 101

Figure 4-21: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 41.21o...... 101

Figure 4-22: Lift coefficient vs. convective time plot for the 0o-85o ramp hold case using the Uhlman method. The results show decreasing CL peaks due to a combination of von Karman shedding and increasing angle of attack...... 103

Figure 4-23: Lift and drag coefficient comparison between the 0o-45o and 0o-85o ramp hold cases showed that the lift coefficients were identical during the ramping motion until the 0o-45o reached its maximum angle of attack...... 105

Figure 4-24: CL vs. angle of attack plot for the 0o-85o ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers and CFD data for a 0.2 span SD7003. Results show a good comparison, though there is more vortex shedding in the code than in the experimental data while it make a good approximation to the CFD data2...... 106

Figure 4-25: CD vs. angle of attack plot for the 0o-85o ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers. Results are good up to 45o, than the DVM code significantly over predicts the

xiv experimental data while it does a good job predicting the CFD data2...... 107

Figure 4-26: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 67.0o...... 108

Figure 4-27: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 75.0o...... 108

Figure 4-28: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 85.0o...... 109

Figure 4-29: CL vs. convective time for the 0o-40o-0o ramp hold case using the Uhlman method...... 110

Figure 4-30: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 20o...... 111

Figure 4-31: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 40o...... 111

Figure 4-32: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 20o...... 112

Figure 4-33: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 0o...... 112

Figure 4-34: CL vs. convective time plot for the ramp hold case showing the influence of each term. It can be seen that the volume integral is the most dominant term in the Uhlman method while the viscous surface integral is negligible...... 114

Figure 4-35: CL vs. convective time plot for the ramp hold case showing the influence of bound vortex number. It can be seen that as n is increased, the stall lift coefficient increased and so did the convective times for vortex shedding...... 115

Figure 4-36: CL vs. convective time plot for the ramp hold case showing the influence of the leading edge separation strength factor. It can be seen that as Ψ is increased, the stall lift coefficient increased slightly while the convective time of vortex shedding shifted as well...... 116

Figure 4-37: CL vs. convective time plot for the ramp hold case showing the influence of the desingularization reduction factor function. It can be seen that the low order algebraic and super Gaussian do not give smooth behavior. All functions give similar trends, with some factors giving a lower or higher slope. The Planetary and Super Algebraic functions give very similar lift coefficients as well...... 118

Figure 4-38: CL vs. convective time plot for the ramp hold case showing the influence of the wake vortex size factor. It can be seen that the size factor did not have significant effect on the lift coefficient...... 119

Figure 4-40: Lift coefficient versus angle of attack for the four reduced frequency values. The fastest reduced frequency case had the lowest slope increase which was most likely caused by the LEV having insufficient time to develop...... 122

Figure 4-43: Lift coefficient versus angle of attack for the four reduced frequency values. As reduced frequency increased, the von Karman shedding was eliminated due to the TEV not rolling up...... 125

Figure 4-44: Drag coefficient versus angle of attack for the four reduced frequency values. As reduced frequency increased, the von Karman shedding was eliminated due to the TEV not rolling up...... 126

Figure 4-45: Convective time versus angle of attack, and lift and drag coefficients...... 127

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Figure 4-47: The velocity profile for the perching case using data from the water tunnel and a curve fit...... 129

Figure 4-49: Angle of attack, lift coefficient, and drag coefficient versus time. After the airfoil stalled, the lift and drag coefficient continued increasing since the dynamic pressure was getting small...... 131

Figure 4-50: Lift per unit span versus time. After the airfoil stalled, the lift began dropping to zero while the von Karman shedding is causing some slight increases in lift...... 132

Figure 4-51: Drag per unit span versus time. The drag force increased post-stall due to the high angle of attack but then dropped due to the decreasing free stream velocity...... 132

Figure 4-52: Velocity, pressure, and vorticity contours at 5 seconds. No vorticity has been developed, and thus no pressure difference to create lift or drag...... 134

Figure 4-53: Velocity, pressure, and vorticity contours at 10 seconds. The LEV has begun forming and has created a low pressure region...... 134

Figure 4-54: Velocity, pressure, and vorticity contours at 15 seconds. The LEV has begun separating from the airfoil surface due to the TEV and shear layer rollup...... 135

Figure 4-55: Velocity, pressure, and vorticity contours at 20 seconds. The second LEV has begun forming while the TEV began separating...... 135

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Figure 4-56: Velocity, pressure, and vorticity contours at 25 seconds. The second LEV continued growing and thus intensifying the low pressure region above the airfoil surface...... 136

Figure 4-57: Velocity, pressure, and vorticity contours at 30 seconds. The airfoil has arrived at 85o and the LEV has begun separating from the airfoil due to TEV and shear layer rollup...... 136

Figure 4-58: Velocity, pressure, and vorticity contours at 33 seconds. The third LEV has begun forming while the shear layer from the TEV flowed vertically along the airfoil’s surface and around the separated LEV...... 137

Figure 4-59: Full flow field plot during the perching maneuver. The vortices did not convect as far due to the decreasing velocity...... 138

Figure 4-60: Flow field comparison between the decreasing free stream and constant free stream case shows that the flow fields did not vary significantly until the airfoil reached high angles of attack, when the velocity was near zero...... 139

Figure 4-61: Flow field comparison between the DVM and experimental results obtained by Ol show a good comparison with some difference, especially at 45o...... 140

Figure 4-62: Velocity contour where the trailing edge shear layer is visible when the airfoil is at an angle of attack of 11.23o...... 142

Figure 4-63: Velocity contour where the trailing edge shear layer is visible when the airfoil is at an angle of attack of 22.46o...... 142

Figure 4-64: PIV and CFD images showing the Kelvin-Helmholtz instabilities that occurred in the shear layer of the 0o-45o ramp-hold canonical case2...... 143

Figure 4-65: Flow field and velocity contour showing the development of the trailing edge shear layer with the growth of the LEV...... 145

Figure 4-66: Flow field and velocity contour showing the development of the leading edge shear layer with the growth of the TEV...... 146

Figure A-1: Example of the red pop-up in MATLAB’s command window to indicate a problem in the coding...... 156

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Figure A-2: Incorrect Wagner function results. Note the dip below the Wagner function before the indicial lift increased to steady state. The results above could not be regenerated in MATLAB, so they were plotted in PowerPoint by memory...... 157

Figure A-3: Corrected Wagner function plot using 100 bound vortices shows strong comparison between DVM and Wagner function approximation...... 158

Figure A-4: Plot showing a vortex shed from the leading edge passing through the airfoil...... 159

Figure A-5: Plot showing a how the vortex is moved from below the airfoil to above the airfoil...... 159

Figure A-6: Plot showing corrected vortex shed from the leading edge without passing through airfoil...... 160

Figure A-7: Increasing angle of attack showed good data until stall, at which the TEV interaction caused bad lift coefficient data...... 161

Figure A-8: Before corrected, the Uhlman method yielded good data only pre-stall. Post- stall, the lift coefficient experienced very anomalous behavior due to the trailing edge roll up during von Karman shedding...... 162

Figure A-9: Lines of code showing the improper location of the TEW_delta_z line...... 162

Figure A-10: Lines of code showing the corrected location of the TEW_delta_z line...... 163

Figure A-11: CL vs. convective time using the Uhlman method after correcting the location of the TEW_delta_z line of code...... 163

xix

LIST OF TABLES

Table 1-1: Various methods implemented in the literature review show a niche in which a low order vortex particle method could be used to analyze high angle of attack and perching-like maneuvers...... 20

Table 3-1: Parameters and test matrix for the steady flow cases...... 48

Table 3-2. Tabular data of the Wagner function given by Garrick23...... 55

Table 3-3. Wagner function flow parameters and test matrix...... 59

Table 3-4. Theodorsen function flow parameters and test matrix...... 62

Table 4-1: 45o impulsive start flow parameters...... 68

Table 4-2: 0o-45o canonical case flow parameters...... 70

Table 4-3: 0o-40o-0o canonical case flow parameters...... 76

Table 4-4: Shedding frequency results for 0o-45o ramp hold case where the physical dimension was the chord...... 94

Table 4-5: Shedding frequency results for 0o-45o ramp hold case where the physical dimension was the projected chord...... 94

Table 4-6: Shedding frequency results for 0o-85o ramp hold case where the physical dimension was the chord...... 104

Table 4-7: Shedding frequency results for 0o-85o ramp hold case where the physical dimension was the projected chord...... 104

xx LIST OF SYMBOLS AND ABBREVIATIONS

α: Angle of attack

αo: Pitch amplitude

: Pitching rate

: Pitching acceleration

γ: Circulation distribution

δ: Flap deflection angle

ρ: Density

ρ∞: Density of free stream fluid

κ: Desingularization reduction factor

θ: Coordinate transformation of ξ into polar coordinate

θo: Coordinate transformation of x into polar coordinate

δ: Wake vortex size factor

ξ: Location of infinitesimal point on the thin airfoil

ξ*: Dimensionless location of infinitesimal point on the thin airfoil

xxi μ: Viscosity

τ: Convective time

: Convective shedding period

: Velocity potential

υ: Kinematic viscosity

ω: Angular velocity

Γ: Circulation

Γj: Jth vortex strength

ΓBound: Total bound circulation

ΓWake: Total wake circulation

ΓTEW: Total trailing edge wake circulation

ΓLEW: Total leading edge wake circulation

ΓTotal: Total circulation

Φ: Non-dimensionalized lift

ΦExact(s): Non-dimensionalized lift using the Wagner function

Φ1(s): Non-dimensionalized lift using the first theoretical approximation to the Wagner function

Φ2(s): Non-dimensionalized lift using the second theoretical approximation to the Wagner function

xxii Ψ: Leading edge separation strength factor

Δp: Pressure difference between the airfoil top and bottom surfaces

Δpi: Pressure difference between the airfoil top and bottom surfaces at ith segment

Δl: Discreet airfoil segment length

Δli: Discreet airfoil segment length of ith segment

Δt: Time interval a: Elastic axis

aij: Influence of vortex j on collocation point i

aiW: Influence of newest shed wake vortex on collocation point i

aiTEW: Influence of newest shed TE vortex on collocation point i

aiLEW: Influence of newest shed LE vortex on collocation point i b: Half chord (c/2) c: Chord

cL: Two dimensional lift coefficient

cD: Two dimensional drag coefficient f: Shedding frequency

ho: Plunging amplitude

xxiii

: Plunging rate

: Plunging acceleration i: Point in which velocity is taken at j: Point at which velocity is taken from k: Reduced frequency n: Bound vortex number

q∞: Dynamic pressure r: Distance between points

rij: Distance between points in ith, jth location in influence matrix s: Two times convective time

: Camberline location t: Time

: Shedding period u: Horizontal velocity

ui: Horizontal velocity at ith collocation point w: Vertical velocity

wi: Vertical velocity at jth collocation point

xxiv uW: Horizontal velocity induced by the wake

wW: Vertical velocity induced by the wake

uiW: Horizontal velocity induced on ith wake vortex

wBound,n: Velocity due to airfoil circulation in the normal direction

wWake,n: Velocity due to wake circulation in the normal direction

wiW: Vertical velocity induced on jth wake vortex x: Horizontal location x*: Dimensionless horizontal location

xi: Horizontal location of ith collocation point

xj: Horizontal location of jth finite vortex

xiW: Horizontal location of ith wake vortex

ΔxiW: Horizontal displacement of ith wake vortex y: Time step counter z: Vertical position

zi: Vertical location of ith collocation point

zj: Vertical location of jth finite vortex

ziW: Vertical location of ith wake vortex

xxv

ΔziW: Vertical displacement of ith wake vortex

Ao: Zeroth order Fourier Coefficient

A1: First order Fourier Coefficient

A2: Second order Fourier Coefficient

An: Nth order Fourier Coefficient

C(k): Theodorsen’s Function

D’: Drag per unit span

G: Green’s function magnitude

H: Enthalpy

Hi: Enthalpy of ith airfoil segment

Ho: Hankel function of the zeroth kind

H1: Hankel function of the first kind

Iij: Influence of enthalpy and Green’s function in the normal direction in the ith, jth location of the influence matrix

L’: Lift per unit span

LE: Leading edge

P: Pressure

Pj: Pressure at jth surface point

xxvi

Pstatic: Static Pressure

Ptotal: Total Pressure

Q: Vertical velocity at the three quarter chord point in complex form

Re: Reynolds number

S: Surface

TE: Trailing edge

U(t): Horizontal kinematic velocity

V: Velocity

Vfluid: Velocity of the fluid

V∞: Free stream velocity

V∞,n: Free stream velocity normal to the airfoil surface

Vi: Known velocity at ith collocation point

: Volume

W(t): Vertical kinematic velocity

Ω: Vorticity vector f: Body force vector n: Normal vector

xxvii ni: Normal vector of ith collocation point

nj: Normal vector of jth airfoil segment

nx: Normal vector x-component

nz: Normal vector z-component u: Velocity vector

FViscous: Viscous force vector

G: Green’s function vector

U∞: Free stream velocity vector

xxviii

CHAPTER 1

INTRODUCTION

1.1 Background

“In general, as for armies you want to strike, the cities you want to attack, and the men you want to assassinate, you must first know the names of the defensive commander, his

assistant, staff, door guards, and attendants. You must have our spies search out and

learn of them” - Sun Tzu, The Art of War

Since the dawn of mankind, people have fought whether it was for food, land, resources, or labor. Warfare has created and destroyed nations, and has been a defining factor in calling some men, such as Gaius Julius Caesar, great. As the centuries passed, weapons and armor have become much more refined, going from bronze to iron, swords to guns, and steel to Kevlar.

Not only have the hand held weaponry changed over, but so have the ground assault vehicles. In ancient times, armies depended on chariots, cavalry, and elephants as their heavy attack platforms. Although chariots and elephants eventually phased out of armies, cavalry always remained. In modern times, horseback cavalry was replaced by armored tanks. However, armies were not only dependent on ground forces. Even in ancient times, navies played a significant role as displayed at the Battles of Salamis and

1 Actium. Battle ships went from triremes to masted frigates to modern destroyers, submarines, and aircraft carriers.

With the creation of the aircraft by the Wright Brothers, this new vehicle was quickly adopted by nations to complement their land and sea forces. Originally Bi- and

Tri-Planes were used in WWI. As the technological capabilities of nations increased, the design of their military aircraft became better. In modern times, military aircraft have become platforms of devastation that in some cases are barely detectable by radar, as demonstrated by the Lockheed Martin F-22 Raptor.

However, all throughout history, warfare has not only been about attack and defense; it has also, and more importantly, been about maneuvering and planning. In order to properly plan, coordinate, and maneuver armies, commanders need to have an understanding of what lies before them on the battlefield (whether that be on a plain, in a city, in the ocean, or in the sky). Therefore, commanders throughout history came up with ways of gaining intelligence over their opponents so that they would have the upper hand in attaining absolute victory.

Commanders throughout history have used spies, double agents, espionage; anything to gain an upper hand on an enemy. For example, George Washington maintained a vast network of spies and double agents in order to gain the upper hand on the British forces during the Revolutionary War. In modern times, militaries began using aircraft to gain reconnaissance. One famous account of this was the Cuban Missile

Crisis. In October of 1962, a U2 was flying over Soviet Cuba when it captured images of nuclear missiles being set up. This led to a great amount of political and military

2 maneuvering by both the United States and Soviet Union, which was brought to an end when the United States agreed to remove missiles from Turkey if the Soviets removed the missiles from Cuba. Without the reconnaissance obtained by the U2, the United States would have been in an incredibly bad position militarily and politically had the missiles in Cuba been activated and armed.

As the decades continued, the United States began developing aircraft that would keep its sons and daughters out of harm’s way. These aircraft became known as

Unmanned Aerial Vehicles (UAVs), which could be flown in regions such as the Middle

East by pilots sitting in a room in the United States. However, many of these UAVs are unable to track targets or gain significant amount of intelligence in urban environments where a major amount of combat is now being fought. Therefore, the United States and several other countries have begun making UAVs smaller into a new class of aircraft known as Micro Aerial Vehicles (MAVs). With the creation of this new, much smaller, division of aircraft, a new class of problems has emerged in the realm of low speed unsteady fluid mechanics, in which two specific examples are flapping wings and perching.

Flapping wings is a concept in which MAV wings perform a motion analogous to a pitching and plunging motion to generate both lift and thrust. The purpose of this is to help MAVs experience animal-like locomotion so that MAVs can maneuver as a bird or bat, as well as experience optimal flight as that experienced by fliers in nature. The difficulty in flapping wings is not only in the physical mechanisms, but also in the very complicated flow fields that the flapping wings generate due to vortex interaction.

Perching, on the other hand, is a high angle of attack pitching maneuver in which the MAV is brought to a near-zero velocity. The purpose of this maneuver is to allow

MAVs to land on surfaces such as a building ledge or tree branch while behaving as if the

MAV belongs in the environment (as a flier in nature would). Aspects of the perching maneuver include significant vortex interaction, high drag, large leading edge vortices, and near-zero landing velocity.

As stated above, the main aspect of the perching maneuver is a high angle of attack pitching maneuver, which in itself is an area of significant research. One group performing research in high angle of attack maneuvers is AIAA Fluid Dynamics

Technical Committee Low Reynolds Number Working Group (FDTC-LRWG). In performing research in this area of interest, two methods are used to analyze the problem: experiments and computational methods. The computational methods can then be broken up into two branches: low order and high order. Low order methods are typically classified as vortex methods (such as vortex particle, vortex lattice, etc.) while high order methods are high fidelity computational fluid dynamics (CFD) analyses.

Although CFD and experiments are very effective analysis tools, they are very temporally expensive methods and are not applicable to rapid-design processes. Low order methods on the other hand are very fast and reliable methods that can be used to help give researchers are general understanding of the flow physics of a specific problem.

For the purposes of the design process, low order methods have the ability to rapidly iterate on various designs. Once a design is selected using the low order methods, it can be sent to the high fidelity CFD and experiments for more accurate analyses.

1.2 Literature Review

As stated in the previous section, there has been a significant amount of research in the area of low Reynolds number unsteady fluid mechanics in both the high angle of attack maneuvers and using low order methods. One of the sets of investigations on high angle of attack aerodynamics was performed by the AIAA Fluid Dynamics Technical

Committee Low Reynolds Number Working Group (FDTC-LRWG). Ol1, et. al., performed experimental and computational analyses on a canonical ramping pitch-up, hold, ramping pitch-down with a maximum angle of attack of 40o. A comparison of the results can be found in Fig. 1-11. All three methods in the figure show a vortex shed from the trailing edge during the ramping pitch-up motion. As the flat plate reaches higher angles of attack, vorticity is shed from the leading edge. As the flat plate pitches down, a counter-rotating vortex gets shed from the trailing edge. As the flat plate finishes the pitch-down maneuver, the shear layer causes the leading edge vortex to detach from the flat plate.

2 with Fig. 1-1, it can be seen that the trailing edge vortex convects further downstream due to the lower pitching rate. The leading edge vortex also grows to be significantly larger than in the case of a higher reduced frequency. As seen in the third row of images, the shear layer grows more and thus leads to higher separation. The separated leading edge vortex convects further in the fourth row while the attached vortex grows.

Figure 1-2: Vorticity plots for 40o ramp-hold-ramp case with reduced frequency k of 0.2 shows a good comparison between Eldredge (first column), Garmann and Visbal (second column), Ol (third column), Lian (fourth column), and Williams (fifth column) 1. Due to the lower pitching rate, the trailing edge vortex convects further downstream and thus reduces its effect on vortex interaction. The pitch rate also allows for the leading edge vortex to grow significantly larger. The shear layer also grows more at a lower reduced frequency. The FDTC-LRWG also studied a second canonical case, the 45o pitch-ramp-hold using an SD7003 airfoil2. Experiments and CFD (as seen in Fig. 4) show instabilities shed from the trailing edge as the airfoil pitched up. During the pitch-up, the leading edge vortex grows and then sheds from its interaction with the shear layer. As the trailing edge vortex grows, the leading edge vortex sheds. As the trailing edge vortex convects downstream, the leading edge vortex reattached. The leading and trailing edge vortices then experience von Karman shedding.

The group found that experimental lift coefficient compared well with the computational lift coefficient once the experimental results were divided by a factor of 1.5 due to blockage. When comparing the lift and drag coefficients between an SD7003 and a flat plate with rounded edges, the group found that lift coefficient (as shown in Fig. 1-42) compared well pre-stall and after the angle of attack reached ~65o. However, the vortex shedding post-stall showed a phase-difference between the SD7003 and the flat plate.

The drag coefficients, on the other hand, compared well over the range of angles of attack.

Figure 1-4: Lift and drag coefficient comparison between an SD7003 and a flat plate for various Reynolds numbers and maximum angle of attack2. The results showed a good comparison, except in the lift coefficient post stall, where there seemed to be a phase difference between the vortex shedding of the airfoil and flat plate. Reich3, et. al., detailed experimental and computational work done on a MAV. A model with four degrees of freedom (two pitching angle rotational degrees of freedom in each wing) were applied to the model and tested in a wing tunnel using fast (10 m/s), medium (7.5 m/s), and slow (5 m/s) wind speeds. Six forces and moments were attained from experimental measurements, in addition to current and voltage readings, and images taken with cameras.

Numerical methods employing 2-d steady, 2-d quasi-steady, steady finite wing, quasi-steady finite wing, and vortex lattice methods were used as a comparison with the experimental data collected for the attached and separated flow regions. Additional time delays were incorporated into the model. The separated flow regions seem to have been modeled well using the methods employed, though it is unclear if the higher order vortex lattice method is required or if a simpler method could be used instead.

The analytical model employed was a time delay model expressed by equation (1-

3 3 1) , where p is a factor that must be solved for from equation (1-2) . Po is a mixing factor

3 defined by equation (1-3) and τ1 and τ2 are factors that can be adjusted to fit the data.

The two dimensional lift coefficients for the attached and separated flow regimes were modeled by equations (1-4)3 and (1-5)3, respectively.

(1-1)

(1-2)

(1-3)

(1-4)

(1-5)

Robertson, et. al.,4 used a similar vortex method as the author of this thesis and applied it to the analyses of high angle of attack aerodynamics and the perching maneuver. He compared his code to flow field results of a 45o impulsively started airfoil and the 0-45o canonical ramp-hold case, as well as comparing lift coefficient results with CFD and experimental results. His code showed a good correlation to other lift coefficient results,

10 except the minimum lift coefficient during the von Karman shedding was significantly lower than minimums of the other methods and seemed to decrease as the von Karman shedding continued. Robertson then went on to detail trajectory optimization using a

Simulink model, as well as showing various trajectories applicable to the perching maneuver, as shown in Fig.

Figure 1-5: Various trajectories for the perching maneuver presented by Robertson4.

Brunton, et. al.,5 used a DNS solver developed at Caltech to produce flow fields around an airfoil at a high angle of attack in unsteady low Reynolds number flow. The authors then went on to describe a low order method involving Galerkin projection onto orthogonal decomposition modes of the Navier-Stokes equation. The low order Galerkin projection and the DNS results showed a good comparison, including the preservation of

Lagrangian coherent structures. The authors also compared the DNS to Theodorsen’s pure plunge and pure pitch simplifications. The pure plunge results showed that the DNS compared well for low Strouhal numbers and up to high reduced frequencies while it did not at high Strouhal numbers and any reduced frequencies. The pure pitching results showed a higher dependence on pitching amplitude and location of the pitching point than on reduced frequency and Strouhal number.

In a related investigation, Brunton, et. al.,6 studied a new model that was more suitable for incorporation into a flight control method. The authors started with presenting a representation of lift coefficient broken into components of average lift coefficient and a component based on state variables. The authors went on to describe how the DNS could not model Theodorsen and Wagner’s functions at angles of attack higher than the stall angle of attack. The Eigensystem Realization Algorithm was then presented, which the authors found to be more efficient than POD models. The results showed a very good comparison between the ERA results and those calculated using

Wagner’s indicial response while the DNS did not provide a good comparison for both pure pitch and pure plunge cases.

Several higher fidelity vortex particle methods were employed by Eldredge, et. al.,7,8. One particle method was a viscous vortex particle method used by Eldredge, et. al.,7 to analyze the canonical pitch up, hold, and pitch down for a flat plate with a reduced frequency of 0.7. The results showed an expected increase in lift and drag during the pitch up. When the angle of attack was held constant, then pitched downward, the lift and drag coefficient data experienced anomalous behavior. The sharp drop in lift and drag coefficient seemed unexpected, though it does make some sense since the new vortex shed from the trailing edge would induce a strong force. An analytical lift coefficient model based on instantaneous angle of attack and reduced frequency was presented to fit the pitch up behavior. The vortex shedding during the pitching motions was then explored.

The analysis of rapidly pitching plate to high angles of attack was again studied by Eldredge, et. al.,8 using both the higher order viscous vortex particle method and a

12 lower order inviscid vortex particle method. Initial investigations were done to determine the role of pitching rate and pitching axis location by using the viscous vortex particle method. By varying the reduced frequency, the lift coefficient results showed an increasing lift coefficient, which then dropped, then increasing to a maximum, which then dropped again. As the reduced frequency was increased, this second peak moved closer and closer to the initial peak until it eventually became a single peak at a reduced frequency of 1. When comparing the lower order particle method with the higher order one, the authors found that the lower order model reproduces the lift and drag time histories, as well as several features exhibited by the higher order code.

Zhang, et. al.,9 used a viscous vortex particle method in conjunction with

Eldredge to analyze the flows around deforming bodies. The mathematics of the method were presented and used on two cases: a submerged cylinder which deforms and a fish like body. The method accurately approximated the analytical solution for the cylinder case as a validation case. The method was then applied to the fish shaped body. The vorticity contours presented showed a von Karman street and a reverse von Karman street depending on the shedding frequency.

A different vortex particle method was used by Monaghan, et. al.,10 to model cyclonic behavior. The authors specifically used the particle method to analyze a

Kirchoff and Kida vortex, which produced accurate results. Vortex images were assembled using a quad tree with smoothed interaction between particles.

Yet another vortex method was implemented by Fishelow11 to analyze viscous fluid flows. The Euler, heat, and Prandtl equations were first discretized, then no-flow and no-slip boundary conditions were enforced. A function was also devised such that

13 given the location of the particles (i.e. whether or not the Prandtl equations are used), it would either act as a blob or a tile. The method was then used to analyze the flow past a semi-infinite plate. Results accurately showed various flow features, such as transition into turbulence.

Two vortex lattice codes were used by Emdad, et. al.,12 to study the ground effect induced on rotor blades using a free wake model. Potential flow theory was briefly mentioned, in addition to the boundary conditions for a rotor blade and free wake (which were broken up into panels) near the ground. The method used was to define the panel locations, enforce the boundary conditions, solve for the vortex strengths, calculate aerodynamic forces and coefficients, adjusting the panels in the wake, and then iterating until a converged solution is found. An initial analysis was first done by comparing an untwisted rotor blade at different rotation speeds using Quasi Vortex Lattice Method to experimental and analytical results, which showed a good comparison. The ground effect analyses were then performed with the QVLM and a Vortex Lattice Method. The coefficients were computed and compared with experiments as shown in Fig. 1-6. The lattice method results did a good job approximating the results.

Figure 1-6: Thrust ratio versus distance from the ground comparisons show that the QVLM and LVM both do a good job approximating the experimental results.12 A discrete vortex method was used by Xu, et. al.,13 to capture flow field effects around an airfoil with a dynamically extended spoiler. The model the authors used to define the strength of the shed vortex from the spoiler tip was using Eq. 1.711, where U is the velocity outside the shear layer and ΔTo is a nondimensional time increment.

(1-7)

Boundary conditions involved the Kutta condition, flow tangency, and the Kelvin condition. Zonal decomposition was used to lower computation time since tracking vortices can be computationally expensive. Two rotation speeds (4000 deg/s and 40,000 deg/s) were studied based on the angle at which the spoiler would end at. Pressure and lift coefficient data was compared with experimental results and showed a good approximation, though the authors did see that the suction peak at the leading edge was overpredicted.

Chabalko, et. al.,14 also used a two-dimensional discrete vortex method to study the flow field induced by a flapping wing. The method was used on a plate (whose camberline matched an SD7003’s) experiencing pure pitch and pure plunge and compared with water tunnel results. A high reduced frequency (3.93) was used in the investigation, whose focus was to optimize the single stroke path parameter. The authors found the optimum at rotation amplitude of 40o. A multi-parameter stroke optimization was then performed and found that the optimum occurred at 70o. However, the authors did find a high level of sensitivity in the optimization process due to small changes in their model parameters.

It was also shown by Murgai, et. al.,15 that the discrete vortex method could be applied to gust responses of separated shear layers. The investigation analyzed the flow around a rectangular block using vortex blobs in the wake and panels on the body’s surface while performing various sensitivity studies on the time step size, number of panels, and blob radius to panel length ratio (RCORE) compared with experimental results. The best values were then used to generate flow fields around the body. The authors found that the mean velocities provided a good comparison to the experimental data while the RMS velocity magnitudes did not (though they did match the trends of the experimental data).

Takeda, et. al.,16 devised four schemes of adding viscous effects into the discrete vortex method: the random walk, diffusion velocity, corrected core-spreading, and vorticity redistribution. In order to reduce the computation time, a fast multipole method was implemented. The methods were then applied to a diffusing Lamb vortex, a diffusing vortex sheet, and an impulsively started cylinder. The authors found that the

16 random walk and diffusion velocity methods converged to the analytical solution for the

Lamb vortex as the number of blobs increased and the vorticity redistribution method gave an excellent comparison while the core spreading method under predicted the vorticity. When comparing the random walk, diffusing velocity, and vorticity redistribution methods with the diffusing vortex sheet, the authors found that the sheet produced by all of the methods except for the vorticity redistribution method broke apart, as shown in Fig. 1-7. In the case of the impulsively started cylinder, the vorticity redistribution method gave the most accurate results.

Figure 1-7: Comparison of four methods on the diffusing vortex sheet: random walk with a seed of 13(a), random walk with a seed of 19 (b), velocity diffusion (c), and vorticity redistribution (d).16

Wickenheiser, et. al.,17 reported an aircraft model (shown in Fig. 1-8) incorporating a blended wing body with rotating outboard wing sections (which allowed for additional degrees of freedom) was used in a study to develop an aircraft that can engage in a perching maneuver, which is a high angle of attack maneuver that uses separated flow and high drag to allow for a planted landing. The aircraft was analyzed using computer programming. The attached flow regime was analyzed using a modified

Wessinger method developed through a lifting line and potential flow theory which incorporated experimental data. In order to optimize the trajectory, both the undershoot

(the amount of distance the aircraft would have to drop below its intended landing site to achieve the proper separated flow to perch) and the distance from the intended landing site required to begin the perching maneuver had to be minimized. For a conventional aircraft design, it was found that a higher thrust-to-weight ratio allows for a lower degree of undershoot. A result for a conventional aircraft design analysis in which the center of gravity was moved from the aircraft’s neutral point was that an unstable aircraft could generate the same pitching moment at slower velocities than other aircraft since it naturally pitches up when perturbed. Analysis was then performed on morphing aircraft.

It was found that by configuring the aircraft to enable high pitching moments, the undershoot of its trajectory was drastically reduced from the conventional aircraft design’s undershoot.

Figure 1-8: Aircraft model employed by Wickenheiser and Garcia that has the ability to morph for the purpose of MAV perching.17 The literature has shown a variety of methods have been used to study unsteady flows where vortex shedding is a very important aspect of the flow physics, as shown in

Table 1-1. Of particular interest is the use of lower order methods to study a variety of flows, whether it be flapping, high amplitude pitching, cyclonic, or viscous flows. The low order methods showed good comparison with experiments and higher order methods in the various investigations. The current investigation seeks to use a similar low order method based on using linear algebraic approximations of differential equations to analyze various fluid flows around an airfoil. This method will be used to study both steady and unsteady flows, low and high angles of attack, and low and high pitching rates. Validation of the code will be made against theoretical, higher order computational, and experimental results. Investigations will include studying the effects of various factors important to the discrete vortex method, such as vortex number. Flow fields and force coefficients will be calculated for these high angle of attack maneuvers

19 where low order methods often do not provide adequate results. Specific high angle of attack cases will be AIAA canonical cases and a perching-like case in which the free stream decreased to zero.

Table 1-1. Various methods implemented in the literature review show a niche in which a low order vortex particle method could be used to analyze high angle of attack and perching-like maneuvers.

Author Method Analysis of Ol Experiment Pitch-ramp-hold-return Garmann/Visbal CFD Pitch-ramp-hold-return Eldredge Viscous Vortex Particle Pitch-ramp-hold-return Method Reich Time Delay Analytical Perching Model Robertson Low Order Vortex Pitch-ramp- Particle Method hold/Perching Brunton DNS High AoA Eldredge Viscous/Inviscid Vortex High AoA Particle Method Zhang Viscous Vortex Particle Deforming bodies Method Monaghan Vortex Particle Method Cyclonic flows Fishelow Vortex Particle Method Viscous fluid flows Emdad Vortex Lattice Method Ground effects Xu Vortex Particle Method Flow around spoiler Chabalko Vortex Particle Method Flapping wing flow fields Murgai Vortex Particle Method Gust responses Takeda Vortex Particle Method Viscous effects Wickenheiser Wessinger Method Perching

CHAPTER 2

THIN AIRFOIL THEORY AND THE DISCRETE VORTEX

METHOD

2.1 Potential Flow Theory and Thin Airfoil Theory

At its most basic level, Thin Airfoil Theory (TAT) is a part of a more general theory in fluid dynamics called Potential Flow Theory (PFT). In low speed aerodynamic applications, there are two primary laws/equations that are used: the Law of Conservation of Mass (2-1 & 2-2)18 and the Law of Conservation of Linear Momentum (2-4 – 2-5)18.

It is worth noting that these equations can be written in Cartesian, polar, or spherical coordinates and throughout this chapter, coordinates will be used interchangeably.

(2-1)

(2-2)

The above expressions in English simply means that the total rate of change of mass in some volume (which is the summation of the mass flow and mass accumulation in ) is equal to zero, where is the “del” or “gradient” operator (2-3)18.

(2-3)

(2-4)

(2-5)

The momentum equations, (2-4)18 and (2-5)18, in English means the total rate of change of momentum is equal to the sum of body, surface, and viscous forces acting on the volume, . By making the assumptions that the fluid is inviscid, incompressible, and irrotational, the Law of Conservation of Mass (LCM) can be reduced to (2-6)18 and (2-

7)18.

0 (2-6)

(2-7)

However, in the above expression, there are three velocity components, which make the analysis complicated. As a simplification, it is useful to postulate that the velocity itself is the rate of change of some potential, , then the velocity components would then be replaced by a single variable in equations (2-8)18.

(2-8)

Then equations (2-6) and (2-7) can be reduced to (2-9)18 and (2-10)18, where (2-9) is known as Laplace’s Equation.

(2-9)

(2-10)

The above equation can then be solved for the potential function, which would then be used to find the velocity by taking the gradient of it.

In fluid mechanics, there exist four elementary solutions to Laplace’s Equation: uniform flow (2-11)18, source flow (2-12)18, doublet flow (2-13)18, and the vortex flow

(2-14)18. Due to the nature of these solutions, the coordinate system will be changed to polar for simplicity. The potential functions for these four solutions are listed below.

(2-11)

(2-12)

(2-13)

Γ (2-14)

Due to the linear nature of Laplace’s Equation, the above solutions can be added together using superposition to model complicated, physical flows. For example, by adding a uniform flow potential and a source potential, one can create a bullet-shaped half-Rankine oval in uniform flow. Similarly, by superimposing a uniform flow potential with a doublet flow potential, a non-rotating cylinder in uniform flow can be modeled.

However, in the case of the current investigation, the superposition of a vortex flow potential and uniform flow potential was of interest. This combination would create the flow around a rotating cylinder, which causes the Magnus effect. By implementing the Zhukovsky mapping technique to correlate an airfoil with a rotating cylinder in a flow, potential flow (via the vortex solution of Laplace’s Equation) could be used to model the flow around the airfoil body.

Now, using this relationship, a finite vortex of strength Γ is placed at the quarter- chord of our airfoil. The vortex itself is broken into a sheet of infinitesimally small vortices of strength γ(ŝ) (illustrated by Fig. 2-118) placed side by side (where ŝ is the coordinate of curvature along the airfoil) and is related to the total vortex strength by equation (2-14)18:

Figure 2-1. Thin airfoil approximation of infinitesimal vortex sheet placed on camber- line18.

(2-14)

Each of the infinitesimal vortices induces some velocity dV on a point in space (where r is the distance from the infinitesimal vortex to the point), related by the equation (2-15)18:

(2-15)

Once the modeled airfoil is placed in a free stream uniform flow (and assuming low speed), the flow has to be streamlined, which means that the velocity vector of the fluid is tangent to the surface of the airfoil, and is shown in Fig 2-2.

Figure 2-2. Streamlined velocity vectors are tangent to the airfoil surface.

This then implies zero fluid velocity normal to the airfoil. From this statement, the primary equation used in the current study can be derived. By knowing that the total fluid velocity normal to the airfoil has to be zero, we know that the summation of the free stream velocity normal to the airfoil and the velocity induced by the infinitesimal vortices normal to the airfoil has to be zero. Therefore, equation (2-16)18 can be written:

(2-16) where the induced velocity by an infinitesimal vortex located at ξ on a location x is given by (2-17)18:

(2-17)

The total induced velocity would then be the sum of the induced velocity by each vortex over the length of the airfoil from the leading edge to the trailing edge. Therefore, (2-17) can be rewritten as (2-18)18:

(2-18)

Once the vortex distribution is found by solving the above equation, the total circulation can be found by using (2-19)18. The lift per unit span (L’) can then be calculated by finding the pressure difference between the upper and lower airfoil surfaces

(2-20)18 and integrating it over the airfoil length (2-21)18. The lift coefficient can then be found with (2-22)18.

(2-19)

(2-20)

(2-21)

(2-22)

Now, time to solve the differential equation. One solution is by using the Glauert method18. By using the following notation transformation found in (2-23)-(2-25)18, (2-

18) can be rewritten as (4-26)18.

(2-23)

(2-24)

(2-25)

(2-26)

After rigorous mathematics, the circulation can be solved for as a function of

18 angle θ, and is described by equation (4-27) , where the constant Ao and An are found using equations (4-28)19 and (4-29)18, respectively. As the summation index, n, approaches infinity, the circulation variation will arrive at the exact solution. To get the circulation distribution in Cartesian x-y coordinates, the inverse of the above transformation would have to be implemented.

  cos1    (2-27)    2   AV o   n sin nA    sin n1 

1  dz (2-28) Ao   do  0 dx

2  dz (2-29) An  cos dn  oo  0 dx

From the eq. (2-3), the pressure difference (2-30)18, lift per unit span (2-31)18, lift coefficient (2-32)18 can be calculated. According to thin airfoil theory, the result for lift coefficient is a function of angle of attack. For a flat plate at low angles of attack, the lift coefficient is (2-33)18. However, for higher angles of attack, equations (2-34)19 can be used for the lift coefficient and subsequent drag coefficient, and will be referenced later on as Flat Plate Theory (FPT), which was a relationship used by Cory and Tedrake19.

(2-30)  Vp   

TE (2-31) L'   pds LE

L L'' (2-32) cL   2 o  AA 1  Sq 1 2   cV )1( 2 

cL  2 (2-33)

(2-34)

As examples, three geometries were considered: a flat plate at an angle of attack, a parabolically cambered airfoil, and a flat plate with a trailing edge flap deflected down. After applying the above method, the circulation distributions were found to be (2-35), (2-36), and (2-37).

(2-35) 1 x     Vx )tan(2 c  x c

1 x 2 c 8a  x   x      Vx  )tan(2       x c  c   c  (2-36) c

1   3 3 3 (2-37)3    V tan2   cot   sin  2sin   4sin   5sin   3  2   2 4 5 

Another method at finding the circulation distribution over the airfoil is by finding the exact solution through the use of the Söhngen inversion formula20. In this case, the airfoil mid-chord is placed at the origin. By non-dimensionalizing the chord length by a factor of c/2 and applying the Söhngen inversion formula

(equations (2-38)20 and (2-39)20, where g(δ*) is known), the nondimensional

circulation variation can be solved for (equation (2-40)20, where w(δ*) is known).

* 1 (2-38) * 1 f   * xg   d 2 1 x   **

* * * 1 (2-39) * 1 1 x 1  g  * xf   d 2 1x* 1 1  * x   **

* * * 1 (2-40) * 1 1 x 1  w  *  x   d 2 1x* 1 1  * x   **

After rigorous mathematics and using a table of integrals, the following solutions

((2-41)-(2-43) were found for the same cases as those performed using the Glauert

method. A comparison of the two solutions will be provided in a later chapter.

1 x* (2-41)  *   Vx )tan(2  1 x*

8aV 2 (2-42)  x*   1 x*  c

      1    1  x*    2 1 x*  1 x*  2    *     (2-43)     Vx   )tan( *  ln    1 x * 2   *   3  1 x  x * 2  3    1 1 x         2 4           

The above derivation applies only to fluid in steady flow. If the airfoil is experiencing unsteady flow, addition terms and equations have to be accounted for. Due to the unsteady motions, vortices get shed off the trailing edge and convect downstream as a vortex sheet. Therefore, the total velocity normal to the airfoil will include a separate contribution due to the wake (2-44)21. It is worth mentioning that the above

29 approximations only apply for a thin airfoil, which means the airfoil’s thickness is much less than the chord length (maximum of 10% of the chord length).

(2-44)

In the case of unsteady flow, an additional equation needs to be added in order to solve for the two unknowns (the bound and wake circulation distributions). This equation is the Kelvin Condition. William Thomson (also known as Lord Kelvin), who is most famous for the unit of temperature named after him (Kelvin), theorized that the total circulation around an object will never change with a change in time, or, as expressed in

Equation (2-45)18,

d  (2-45)  0 dt

For example, an airfoil initially at rest (Fig. 2-3a)18 will have an initial circulation of zero. Once a free stream of velocity V∞ begins at time t1, according to Kelvin’s

Theorem of Circulation, the circulation will not change and therefore will be zero.

Figure 2-3. Diagram of the circulation around an airfoil when airfoil is at rest and at the beginning of forward motion18.

However, since we also know that circulation is related to velocity (the details of which will be explained in the following section), there has to exist circulation around the airfoil even though Kelvin’s theorem says there should be zero circulation. The solution to this paradox is found by breaking the total circulation, ΓTotal, into two regions, one around the airfoil called the bounded circulation and one behind the trailing edge called the wake circulation. The total circulation will remain zero, but the sub-circulations will each have a value (equation (2-46)21).

Total 0 Bound  Wake (2-46)

In fact, one finds that the circulation in the bounded and wake regions rotate in opposite directions (as seen in equation (2-46) and Fig. 2-3b18); i.e., if the bounded circulation is positive, the wake circulation is negative. (2-46) can then be rewritten as

(2-47)21.

Wake  Bound (2-47)

This is verified in experiments that show a vortex generated at the trailing edge with counter-clockwise rotation. As the bound vorticity increases, the rear stagnation point moves towards the trailing edge. Once the rear stagnation point reaches the trailing edge, the velocity and vorticity above and below the airfoil at the trailing edge will be equal.

This occurrence is called the Kutta condition. When the Kutta condition is satisfied, the wake vortex will break away from the airfoil and flow downstream to infinity, where it concentrates into a single vortex called the starting vortex.

Now, the above equations and assumptions apply only to airfoils at low angles of attack (α < 10o). Experiments have shown that as an airfoil goes to higher angles of

31 attack, the flow actually separates at (or near) the leading edge. This is due to the stagnation point which has moved from the leading edge to a region on the bottom surface of an airfoil. When the fluid encounters the stagnation point, it either continues moving down the airfoil or it rolls over the leading edge. At high angles of attack, the fluid is unable to stay attached to the airfoil surface and thus separates from the surface, creating a region of vorticity, as shown in Fig. 2-421.

Figure 2-4. Flow remains attached at low angles of attack; separates at high angles of attack when the fluid is forced to wrap around the leading edge.21

When this occurs, vorticity sheds and convects downstream. These vortices also impart velocity on every point in the flow, including shed regions of vorticity. So therefore, the velocity induced by these vortices has to be taken into account by our equations (2-48). The first basic equation relating the induced velocity to free stream must now be rewritten to incorporate these new vortices. In addition, the Kelvin conditions must also be updated. However, now there are N+1 equations and N+2 unknowns (2-49). The solution to this problem will be addressed in the next chapter.

(2-48)

Total 0 Bound TEW  LEW (2-49)

This section provided the theoretical background of potential flow theory and its application to thin airfoil theory. The basic equations of steady and unsteady aerodynamics were introduced, as well as mentioning the addition of vortices shed from the leading edge. It is these equations that will be implemented into the core of the thesis work, which is the Discrete Vortex Method.

2.2 Discrete Vortex Method

Now that the underlying concepts of aerodynamics, potential flow theory, and thin airfoil theory have been firmly established, discussion can be moved to topic of interest: the Discrete Vortex Method (DVM). In the realm of low speed, unsteady aerodynamics, there exist a multitude of complicated problems, which include flapping wings and perching maneuvers. There is also a multitude of methods by which to solve these problems, which include experimental methods, computational fluid dynamics

(CFD) methods, vortex lattice, vortex panel, source panel, etc.

However, a number of these methods are computationally expensive and may require days, weeks, or even months for single tests to be completed. It is then desirable to have a fast, accurate, and powerful method that can allow for an iterative process during the design process of a vehicle that gives strong approximations to what would be found in a full CFD solution. In the case of this thesis, the DVM was the method selected for the task of analyzing a sub-class of problems encountered in unsteady aerodynamics.

What then is the DVM? The DVM is an approximation of Thin Airfoil Theory that represents the continuous vortex sheet along an airfoil and in the wake as a series of point vortices. In the current implementation of the DVM, a two dimensional approximation is used in which the airfoil is broken into n number of discrete segments of length Δl (where Δl = c/n). Each segment then has a vortex of finite strength and size

(whose importance will be mentioned later) located at its quarter chord and a collocation point located at its three-quarter chord where the flow tangency boundary condition is enforced (as shown in Fig. 2-521).

Figure 2-5. Airfoil shape plot with the finite vortex and collocation points shown for n = 7.21

With the basic concept of the DVM laid out, the mathematics of the method will be discussed. The derivation behind the DVM begins with the zero normal velocity boundary condition, eq. (2-50)18.

(2-50)

In the case of theory, the velocity induced by an infinitesimal vortex, dV, by some point x on point ξ was written as eq. (2-51)18:

(2-51)

By dealing with finite vortices instead of infinitesimal vortices, the differential expression can be replaced by a numerical equation as follows (2-52)21:

(2-52)

where the total distance between the two points is given by (2-53)21:

(2-53)

The above equations say that the combination of the horizontal and vertical components of velocity (u and w, respectively) on collocation point i induced by vortex j is equal to the strength of the vortex times the x- and z- distance between the two points divided by 2π times the total distance between the points. Taking the product dot of this influence for a unit strength vortex in the direction of the normal vector gives the influence coefficient of the jth vortex on the ith collocation point, expressed by (2-54)21.

In order to get the free stream velocity component normal to the airfoil, we must take the dot product between the normal vector of the collocation point and the free stream velocity vector to get its influence on the collocation points (2-55)21.

(2-54)

(2-55)

Therefore, the total velocity at the collocation point i can be written as equation

(2-56)21, where the normal velocity induced by the free stream is added to the velocity induced by the finite vortex, which is multiplied by its influence coefficient at the collocation point.

(2-56)

By taking the summation of the velocity induced by every discrete vortex on the collocation point i, in this case point 1, we can generate equation (2-57)21.

(2-57)

Performing the same analysis on every collocation, we can generate n equations for n unknowns (the finite vortex strengths). By putting the equations into a matrix, equation

(2-58) can be generated.

(2-58)

Therefore, the strength of each finite vortex would be found by taking the inverse of the influence matrix and multiplying it by the normal velocity vector, eq. (2-59).

(2-59)

The pressure difference (2-60)21, lift per unit span (2-61)21, and lift coefficient (2-

62)21 can be then computed with the following equations. It is interesting to note that the

Kutta-Joukowski equation falls out of the expression for lift per unit span.

(2-60)

(2-61)

(2-62)

In order to incorporate the transient aspect of thin airfoil theory into the analysis, unsteady aerodynamics of Thin Airfoil Theory had to be understood. Unsteady aerodynamics is concerned with the aerodynamics of an airfoil experiencing dynamic variation that changes with time (which could come in the form of a pitching airfoil to some angle of attack α, an airfoil accelerating from some initial free stream velocity, V∞,i, to a higher or lower free stream velocity, V∞,f, an airfoil plunging (translating in the vertical direction), etc.).

For unsteady aerodynamics, equation (2-50) can be modified to incorporate a wake developed at the airfoil trailing edge, shown by Fig. 2-621.

Figure 2-6. Airfoil shape plot which incorporates vortices shed from trailing edge21.

This is easily done by adding to equation (2-50) the velocity induced by the wake vortex normal to the airfoil’s surface and is described by equation (2-63).

(2-63)

However, that leaves one equation and two unknowns (the vortex distribution strength of the airfoil and the vortex distribution strength of the wake). How then do aerodynamicists solve for these two unknowns? The answer is by using the Kelvin condition, which states that the time rate of change of circulation of a system does not

21 change with time and is expressed by equation (2-64) , where ΓBound(t) is the total airfoil circulation at time t and ΓWake is the total circulation shed into the wake.

(2-64)

By solving these two simultaneous equations, both unknowns can be found, but only for a given instant in time. If the properties/orientations of the fluid or airfoil are changing with time (which means the circulation of the airfoil and wake are also changing with time), the vortex strengths of the airfoil and wake have to be recalculated for each given instant in time.

This can be done by using the DVM developed earlier, with modifications to take into account the unsteady aerodynamics. The unsteady DVM now incorporates the wake vortices and the Kelvin condition, broken into time steps, where ti= ti-1+Δt. At any time- step y, equation (2-59) is satisfied at the collocation points of each discrete segment.

That will create n number of equations for n+1 unknowns (Γk, where k = 1, 2, 3…n, and

ΓW at time-step y).

By then employing the Kelvin condition, there will result N+1 equations and N+1 unknowns. If wake vortices exist from previous time-steps, the velocity induced by them at the collocation points must also be taken into account. The Helmholtz condition for vortex decay states that the strength of a vortex shed into the wake of an object does not

38 decay with time. Thus, the strength of the pre-existing wake vortices are known. Now,

N+1 equations remain with N+1 unknowns for each time-step and can subsequently be used to calculate the strength at each discrete vortex and wake vortex for each time-step.

The pressure difference, lift, and pitching moments can now be calculated (incorporating unsteady effects) at each time-step.

By using (2-52) and (2-53) for the wake vortices, the wake influence coefficient

(eq. (2-65)21) for the current time step wake and the influence of previous wakes can be calculated. With the influence of previous wakes found using equation (2-52), the velocity induced by the unsteady motion, V(t), can be calculated (expressed by equation

(2-66)21). In equation (2-67)21, the normal velocity boundary condition is being enforced at the ith collocation point.

(2-65)

(2-66)

(2-67)

By taking the summation of the velocity induced by every discrete vortex on the collocation point i, in this case point 1, equation (2-68)21 can be generated.

(2-68)

When the normal boundary condition is enforced at each collocation point, the simultaneous equations can be put into matrix form (eq. (2-69)21), where the N+1 row defines the Kelvin condition.

(2-69)

Equation (2-69) can be re-written using linear algebra into equation (2-70) to solve for each bound vortex strength and the strength of the wake vortex shed from the trailing edge of the current time-step.

(2-71)

The pressure difference for each discrete segment can be computed using equation (2-

72)21 while the lift per unit span can be calculated with equation (2-61).

(2-72)

In addition, the shape of the wake (wake rollup) can be calculated using equation (2-73)21 and equation (2-74)21.

(2-73)

(2-74)

However, the above equations are only applicable for small angles of attack (i.e., no leading edge separation). If leading edge separation occurs, the equations above have to be modified to include the leading edge separation vortices, as shown in Fig. 2-721.

Figure 2-7. Airfoil shape plot which incorporates vortices shed from leading edge21.

By incorporating the influence of the leading edge vortices on the collocation points, there will exist N equations with N+1 unknowns. One method for determining the strength of the leading edge vortices was by using a method employed by Robertson, et. al.,4 in which he kept the vorticity per unit length constant. By using a similar equation, the additional equation used was (2-75)4, where the coefficient Ψ is a strength factor based on the segment length, time step size, and free stream velocity. By setting the strength factor (2-76)4, the required time step size can be found.

(2-75)

(2-76)

The result of this approximation allows N+2 equations for N+2 unknowns. It will be shown in the results chapter of this thesis that using the above approximation gives very good results for the LE wake shapes. By taking the summation of the velocity induced by every discrete vortex on the collocation point i, in this case point 1, the equation (2-77)21 can be generated.

(2-77)

When the normal boundary condition is enforced at each collocation point, the simultaneous equations can be put into matrix form (equation (2-78)21), where the N+1 row defines the Kelvin condition and the N+2 row is equation (2-75).

(2-78)

Equation (2-78) can be re-written using linear algebra into equation (2-79) to solve for the discreet and wake circulation (LE and TE) of the current time-step.

(2-79)

It is worth noting that the influence and locations of the LE vortices are found in the same way as was used to find the location and influence of the TE vortices. The problem occurs when the code attempts to calculate the pressure (and thus lift) using the unsteady

Bernoulli equation. If that equation is used without modification to take into account

“unbound” vorticity when LE separation exists, the resulting pressure differences are completely wrong, as will be shown in the results section.

The equations described in this section were then written in a code using the program MATLAB. The code itself was broken into a series of scripts, as detailed by the flowchart of Fig 2-8. The code activates by running the code implementation script in the

MATLAB command window.

Code Implementation Script Input Parameters

Unsteady Flow Parameters

Location of Leading Edge

Airfoil Segment Generator

Timesteps begin

Airfoil Shape Generator

Vortex, Collocation Point, Wake, and Surface Point Location Creators

Influence Matrix Creator and Inverter

Wake Rollup

Pressure Calculator

Force Calculator

Post Processing

Figure 2-8. Flow chart of DVM code

As mentioned at the beginning of this chapter, each vortex is modeled with a finite size. It will be seen in the results section that this finite size plays a crucial role in vortex interaction when circulation is shed from the leading edge. In the code, the bound and wake vortices were not modeled as points but as “blobs” with a finite radius. This was to allow for the modeling of the vortices shed from the leading edge, which could not have been done without giving the vortices a finite radius.

When vortices enter this radius, their influence is reduced according to a desingularization function22 so that singularities do not occur. The influence reduction factor, κ, for several desingularization functions were plotted against the distance between the vortices and point in question in Fig. 2-9. By implementing the reduction factor, equation (2-80) can be rewritten as (2-80). In addition, the vortices shed into the wake had radii defined as factor, δ, multiplied by the radius of the bound vortices (2-81).

By incorporating the size of the vortices and desingularization reduction function, the code pseudo-modeled viscous effects when vortices got close to one another.

(2-80)

(2-81)

Reduction Factor vs. p 1.4

1.2 

0.8 Planetary Model Low Order Algebraic 0.6 High Order Algebraic Gaussian Smoothing Super-Algebraic Smoothing 0.4 Super-Gaussian Smoothing Induced Velocity Reduction Factor, Factor, Reduction Velocity Induced 0.2

0 0 1 2 3 4 5 6 Distance Between Points/Vortex Radius Figure 2-9. Various desingularization reduction factor plots show how the vortex influenced is reduced as vortices get close to each other22.

What sets this method apart from Vortex Lattice, Vortex Panel, or CFD approaches is that this is a much simpler representation, and thus does not require the high computational time of the other methods. The drawbacks to this implementation of the DVM are that it is only applicable to a thin airfoil and is two-dimensional instead of three dimensional (which incorporates wing tip effects due to the roll up vortices and cross-flow). The low order nature of the technique also means that it will not model the flow field as accurately as CFD or experiments.

Due to inherent unreliability when using numerical methods, some sort of validation has to be made in order to verify the results of the numerical method. This validation can come in the form of comparing the numerical results to experiments, theory, or other computational work. In addition, as more vortices are shed into the

45 wake, the computation time increased a lot. Even with drawbacks, the DVM is nonetheless a powerful tool for the analysis of aerodynamics on an airfoil.

CHAPTER 3

STEADY FLOW AND LOW ANGLE OF ATTACK UNSTEADY

AERODYNAMIC VALIDATION

3.1 Steady Flow Validation

Once the steady flow code was written, it required validation. The validation was performed by plotting the circulation distribution plots and comparing them with theoretical solutions (Söhngen and Glauert, specifically). For the analyses, a test matrix

(Table 3-1) was generated to incorporate the three cases that were analyzed (a flat plate, a parabolically cambered airfoil, and flat plate with trailing edge flap). The resulting circulation distribution plots for the three test cases can be found in Fig. 3-1-3-3.

Table 3-1: Parameters and test matrix for the steady flow cases

3 V∞ (m/s) c (m) ρ∞ (kg/m ) μ∞ (Pa-s) Re

1 1 1 1.00E-04 1.00E+04

Test Matrix

n 10 100 1000

Cases Case 1 Case 2 Case 3

α 5o 0o 0o

z(x) Flat Plate Parabolic Flat Plate with TE Flap

δ N/A N/A 6.5o

As can be observed in comparing the five curves in Fig. 3-2, the circulation at leading edge does approximate the exact solution (2-32) at higher numbers of discrete vortices (100 and 1000). In addition, after the location of approximately -0.375 on the x- axis, all five discrete vortex curves overlap. The circulation distribution is equal to zero at the trailing edge for all five curves, showing that the Kutta condition was enforced. It is worth mentioning that the above figures use an unconventional notation, where the mid-chord is placed at the origin. The reason for this new convention was because the

Söhngen Inversion Formula was solved with the mid-chord being placed at the origin.

z(x) vs. x,  = 50 Airfoil 0.05

Airfoil 0 z(x)

-0.05 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x (x) vs. x,  =5o 2  n=10 (x)

 1.5  n=100 1  n=1000  0.5 Glauert

Circulation, Circulation,  Sohngen 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x

Figure 3-1: Shape and circulation curves for the three test runs on a flat plate at 5o angle of attack to display the strong approximation when compared to the results using the Söhngen Inversion Method and Glauert Method. The coordinate system was shifted to the mid-chord since the Söhngen Inversion Method results were calculated with the mid- chord at the coordinate system origin. From Fig. 3-2, the profile of the circulation distribution resembles the shape of the camber, which is parabolic. The circulation is a maximum at the mid-chord and minimum at the leading and trailing edges (zero, enforcing the Kutta condition at the TE).

The DVM therefore very strongly approximates the theoretical solutions, further validating it in steady flow.

z(x) vs. x,  = 00 Airfoil 0.05

Airfoil -0.05 z(x) -0.1

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x (x) vs. x,  =0o 0.8  n=10 (x)

 0.6  n=100  0.4 n=1000  0.2 Sohngen  Circulation, Circulation, Glauert 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x

Figure 3-2: Shape and circulation curves for the three test runs on a 10% cambered airfoil at 0o angle of attack to display the strong approximation when compared to the results using the Söhngen Inversion Method and Glauert Method. The coordinate system was shifted to the mid-chord since the Söhngen Inversion Method results were calculated with the mid-chord at the coordinate system origin. As can be seen in Fig. 3-3, the airfoil experienced a singularity at the leading edge, which is similar to the flat plate at an angle of attack. This leads to the conclusion that the flap deflection creates an effective angle of attack by generating camber. The code matched the singularity at the leading edge, as well as the singularity which occurred at the hinge point. The Glauert solution did not model this singularity, which means that more Fourier coefficients were required in the solution since only five were used in the derivation. This last test case gave the final validation required for steady flow, allowing for the validation of the code for unsteady flow cases.

z(x) vs. x,  = 0o &  = 6.5o Airfoil 0.05

Airfoil 0 z(x)

-0.05 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x

(x) vs. x,  =0o &  = 6.5o 0.6 

(x) n=10   0.4 n=100  n=1000 0.2  Sohngen

Circulation, Circulation,  0 Glauert -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x

Figure 3-3: Shape and circulation curves for the three test runs on a flat plate with a trailing edge flap deflected down 6.5o to display the strong approximation when compared to the results using the Söhngen Inversion Method and Glauert Method. (Note the singularities at the leading edge and hinge location). The coordinate system was shifted to the mid-chord since the Söhngen Inversion Method results were calculated with the mid-chord at the coordinate system origin. With the code written for steady aerodynamics, three geometries were analyzed to determine how accurate the results could be, as well as determine the dependence of bound vortex number. The results showed that the code did a good comparison in steady flow theoretical results for the various geometries. Since for unsteady cases 1000 vortices is too expensive, the bound vortex number will be kept at 100 or less for future analyses.

3.2 Classical Unsteady Aerodynamics

In the area of unsteady aerodynamics, there exist two primary classical functions developed in the early 1900’s that define the lift generated by a thin airfoil experiencing unsteady motion: the Wagner function and Theodorsen’s function. Matching the

51 theoretical results of these two functions will be the validation for the code’s unsteady code while implementing similar assumptions used by both Wagner and Theodorsen: thin airfoil, inviscid, incompressible flow with small oscillations and angles of attack.

The Wagner function is a curve which details, according to Garrick23, “the growth of circulation or lift about an airfoil at a small fixed angle of attack starting impulsively

23 from rest to a uniform velocity v.” The Wagner function (ΦWagner(s)) is a non- dimensional number that models the asymptotic increase in lift due to the decreasing influence of the shed starting vortex and corresponds with the transient lift divided by the steady-state lift, expressed by equation (3-1). Figure 3-423 is a plot of the Wagner function with respect to s, defined as twice the convective time, where convective time is defined by (3-2)20.

tL )( (3-1)  tL  )(

tV (3-2)    c

Figure 3-4. The Wagner function plot versus s23. Note, Garrick defines the Wagner function using k, which in this manuscript is reduced frequency. As can be seen in Fig. 3-4, the Wagner function starts off at 0.5 (one half the steady value). As time progresses, the non-dimensional lift increases gradually until it arrives at the steady value (1). The transient lift for the above approximations would then be calculated using equation (3-3)23.

(3-3) Wagner  2 bL  wVWagner s)(

Katz, et. al.,20 employed the DVM using a single discrete vortex (Lumped Vortex

Method ) to approximate the Wagner Function. The method did not approximate the non-dimensional lift well at low convective times. As can be seen in Fig. 3-520, the non- dimensional lift approximation simply curved down from the singularity near t = 0 to the steady value instead of dropping first to 0.5. Katz and Plotkin also plotted the wake shape resulting from the impulsively started airfoil, which can be seen in Fig. 3-620. The rollup of the starting vortex (located at just before x = 0) is clearly visible in the figure.

Figure 3-5. A comparison between the Wagner Function and the Lumped Vortex Method shows a very poor approximation at low convective times21.

Figure 3-6. The wake shape generated from an impulsively started airfoil21.

20 Two approximations of the Wagner function were defined by Bisplinghoff , Φ

1(s), expressed in equation (3-4) and Φ 2(s) defined by equation (3-5). A comparison of these two approximations compared to the exact Wagner function (as plotted using tabular data from Garrick, Table 3-2.) can be found in Fig. 3-7. It can be clearly seen that the second approximation (which involves exponential functions) does a much better job

54 approximating the data in Table 3-2. However, both approximations will be used in the validation study.

2 (3-4) s 1)(  1 4  s

 .0 0455s  3.0 s (3-5) 2 s  .01)( 165e  .0 335e

Table 3-2. Tabular data of the Wagner function given by Garrick23.

s ΦExact(s) -1.0 0.000 -0.5 0.000 0.0 0.500 0.5 0.556 1.0 0.601 2.0 0.669 3.0 0.720 4.0 0.758 5.0 0.788 6.0 0.813 7.0 0.833 8.0 0.849 9.0 0.863 10.0 0.875 20.0 0.932

Wagner Function vs. Approximations

0.8

 Exact  0.6 Approximation 1

  Approximation 2

0.4

0.2

0 0 2 4 6 8 10 12 14 16 18 20 U t/c o

According to Garrick23, Theodorsen’s function details “the lift on an airfoil oscillating sinusoidally through a small angle of attack and moving with uniform velocity v”, and is described by equation (3-6)23, where C(k) and Q are defined by equations (3-7)20 and (3-

20 20 20 8) , respectively, and H1 and H0 are Hankel functions.

   )(2 QkCVbL (3-6)

)2( (3-7) H1 kC )(  )2( )2( H1  iH 0

Q  we ti (3-8)

It is obvious by looking at (3-7) and (3-8) that the lift on the airfoil will have both a real and imaginary component. By taking the real component, one would eventually arrive at the expression found in equation (3-9)20 for lift, which takes into account the

56 various types of oscillatory motions that can be experienced by the airfoil. In the equation, the term on the right corresponds to the circulatory component while the left term is the added mass/inertial component.

 1  (3-9)       LTheodorsen   Uhb ba  2VbC     abUhk     2  

The sinusoidal oscillatory motion of the airfoil can come in three forms: pure plunging, pure pitching, and combined pitching and plunging (where plunging is vertical displacement and pitching is rotation about a defined point). The motion of the plunging and pitching are sinusoidal, defined by equations (3-10)20 and (3-11)20, respectively.

 o thth )sin()( (3-10)

t  o  t)sin()( (3-11)

In the above equations, h or α is the amplitude of the motion and ω is the angular velocity of the motion. The angular velocity in the above expressions can be found by rearranging the expression for a very important non-dimensional parameter in unsteady aerodynamics called the reduced frequency (k), defined by equation (3-12)1.

c (3-12) k  2V

A comparison of Theodorsen’s function for pure plunging airfoil (in which h =

0.019*c) can be found in Fig. 3-8 for three reduced frequencies: 1.0, 0.5, and 0.1. A similar comparison for a purely pitching airfoil can be found in Fig. 3-9, in which the

o amplitude is αo= 5.5 .

L(t) vs. U t/c, h = 0.2 o o 0.1

0.05 L Theodorsen, k=1.0 L Theodorsen, k=0.5 0

L(t) L Theodorsen, k=0.1

-0.05

-0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 U t/c o h(t) 0.02

0.01 h(t) k=1.0 0 h(t)

h(t) k=0.5 h(t) k=0.1 -0.01

-0.02 0 0.5 1 1.5 2 2.5 3 3.5 4 U t/c o

Figure 3-8. Theodorsen’s function for a purely plunging airfoil at k = 1.0, 0.5, and 0.1.

L(t) vs. U t/c,  = 5.50 o 0.1

0.05 L Theodorsen, k=1.0 L 0 Theodorsen, k=0.5 L(t) L Theodorsen, k=0.1 -0.05

-0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 U t/c o (t) 0.02

0.01 (t) , k=1.0

(t) 0 (t)

 , k=0.5 (t) -0.01 , k=0.1

-0.02 0 0.5 1 1.5 2 2.5 3 3.5 4 U t/c o

Figure 3-9. Theodorsen’s Function for a purely pitching airfoil at k = 1.0, 0.5, and 0.1.

The above section describes two theoretical lift functions, the Wagner function and Theodorsen’s function, commonly applied to four types of flows: impulsive start, pure pitching, pure plunging, and combined pitching/plunging. It is these functions that the DVM code will be compared to as validation in the low amplitude cases.

3.3 Unsteady Flow Validation

With the steady code validated by comparing it with circulation distributions for various thin airfoil geometries, the code was then validated against two unsteady functions: the Wagner function and Theodorsen’s function. The first validation case compared the code to the Wagner function for an impulsively started airfoil at a low angle of attack. A test matrix (Table 3-3) was generated, describing the various parametric studies performed to observe the sensitivity of discrete vortex number and time step size.

Table 3-3: Wagner function flow parameters and test matrix.

3 V∞ (m/s) c (m) ρ∞ (kg/m ) μ∞ (Pa-s)

1 1 1 1.00E-04

Re α z(x) t (s)

1.00E+04 5o Flat Plate 4

Test Matrix

n 1 10 100

Δt (s) 1 0.1 0.01

The first parametric study performed involved varying the value of n and keeping the time step size fixed (0.01s). The results were plotted in Fig. 3-10. The results in the figure showed a very good comparison between the DVM approximation and the theoretical approximations when n = 100. At higher convective times, the approximation when n = 10 also gave a satisfactory comparison to the theoretical approximations.

However, when n = 1, the comparison was very poor. In addition, the second theoretical approximation seemed to be the better approximation to the exact Wagner function.

 vs. U t/c o 1.1

 n=1, t=0.01 s 0.9  n=10, t=0.01 s  n=100, t=0.01 s 0.8  1

  2 0.7 Asymptote

0.6

0.5

0 0.5 1 1.5 2 2.5 3 U t/c o

Figure 3-10: The indicial lift curves with Δt of 0.01 seconds for 1, 10, and 100 discrete vortices compared to the two Wagner approximations shows a very strong agreement between the two theoretical approximations and the result when n = 100. The second parametric study performed involved varying the time step size and keeping the value of n fixed (100). The results were plotted in Fig. 3-11. The results in the figure showed a very good comparison between the DVM approximation and the theoretical approximations when Δt = 0.1s and 0.01s. However, as expected, when Δt =

1s, the comparison was poor.

 vs. U t/c 1.1 o

 0.9 n=100, t=1 s  n=100, t=0.1 s  0.8 n=100, t=0.01 s   1  0.7 2 Asymptote

0.6

0.5

0 0.5 1 1.5 2 2.5 3 U t/c o

Figure 3-11: The indicial lift curves with n = 100 for a Δt of 1, 0.1, and 0.01 compared to the two Wagner approximations shows a very strong agreement between the two theoretical approximations and the result when Δt = 0.1 and 0.01s. Several conclusions can be drawn from Figs. 3-10 and 3-11. The first conclusion is that the case using 100 vortices approximated the exact solution very well for the Δt of

0.1s and 0.01s. The second conclusion is that as the time-step interval Δt goes to zero and the number of vortices goes to infinity, the approximation would approach the exact solution for the Wagner function.

The unsteady code succeeded in approximating the Wagner function and the resulting wake for a flat plate at a low angle of attack impulsively started from rest to the free stream velocity. The next important validation would be to approximate

Theodorsen’s function for a purely plunging and a purely pitching airfoil. The test matrix for the study can be found in Table 3-4.

Table 3-4: Theodorsen function flow parameters and test matrix.

3 V∞ (m/s) c (m) ρ∞ (kg/m ) μ∞ (Pa-s)

1 1 1 1.00E-04

Re Δt (s) t (s) z(x)

1.00E+04 0.01 4 Flat Plate

Test Matrix

n 1 10 100

k 0.1 0.5 1

Cases α o h o

Pure Plunge 0o 0.05c

Pure Pitch 5o 0c

The first parametric study performed was on a flat plate experiencing pure plunge.

The resulting transient lift curves and wake shapes were then plotted against

Theodorsen’s function in Fig. 3-12. When the reduced frequency was 0.1, the approximation matched very well to Theodorsen’s function for every value of n.

Although it may seem that nothing happens in the 0.1 plot due to scaling, the results were re-scaled in Fig. 3-13 so that the sinusoidal shape of the lift could be seen.

When the reduced frequency was 0.5, all three approximations compared well to

Theodorsen’s function, however approximation disagreement between n values started to become noticeable. When the reduced frequency was 1.0, the approximations when n =

10 and 100 provided good comparisons while the approximation when n = 1 did not. It

62 should be mentioned that since Theodorsen’s function itself is a mid-plane approximation, the DVM solution most likely yielded a result more comparable to the exact, physical solution. In addition, the wake shape took on a more sinusoidal shape as the reduced frequency was increased, showing the von Karman alternating vortex street off the trailing edge. It is worth mentioning that the scaling on the lift for the three reduced frequencies was the same. This was to show that at low reduced frequencies, not much happened. As the reduced frequency increased, the lift became more sinusoidal.

L(t) vs. U t/c, k = 0.1, h = +/- .05*c z(x) vs. x, k = 0.1, h = +/- .05*c o o o 0.5 0.5 L n=1 L n=10 0 0 L(t)

L z(x) n=100 L Theodorsen -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x L(t) vs. U t/c, k = 0.5, h = +/- .05*c z(x) vs. x, k = 0.5, h = +/- .05*c o o o 0.5 L 0.5 n=1 L n=10 0 0 L(t)

L z(x) n=100 L Theodorsen -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x L(t) vs. U t/c, k = 1.0, h = +/- .05*c z(x) vs. x, k = 1.0, h = +/- .05*c o o o 0.5 0.4 L n=1 0.2 L 0 n=10 0 L(t) L z(x) n=100 -0.2 L -0.5 Theodorsen -0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c x o

L(t) vs. U t/c, k = 0.1, h = +/- .05*c o o 0 L n=1 L -0.005 n=10 L n=100 L Theodorsen -0.01

-0.015

-0.02 L(t)

-0.025

-0.03

-0.035

-0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 U t/c o

Figure 3-13: Re-scaled lift curves compared to Theodorsen’s function for pure plunging at k = 0.1 to show that the airfoil did experience transient lift at the low reduced frequency. The second case to compare with Theodorsen’s function was a flat plate in pure pitch experiencing the same conditions and reduced frequencies as the pure plunge case.

The lift results and wake shapes can be found in Fig. 3-14. Similar conclusions as those in the pure plunging case can be drawn from Fig. 3-14.

o o L(t) vs. U t/c, k = 0.1,  = +/- 5 z(x) vs. x, k = 0.1,  = +/- 5 o o o 0.5 0.5 L n=1 L n=10 0 0 L(t) L n100 z(x) L Theodorsen -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x o o L(t) vs. U t/c, k = 0.5,  = +/- 5 z(x) vs. x, k = 0.5,  = +/- 5 o o o 0.5 1 L n=1 L 0.5 0 n=10 L(t)

L z(x) n100 0 L -0.5 Theodorsen -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x o o L(t) vs. U t/c, k = 1.0,  = +/- 5 z(x) vs. x, k = 1.0,  = +/- 5 o o o 0.5 0.5 L n=1 L 0 n=10 0 L(t) L z(x) n=100 L Theodorsen -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x

Figure 3-14: Transient lift curves compared to Theodorsen’s function and wake shapes for pure pitch at k = 1.0, 0.5, and 0.1 show very good comparison between the DVM and Theodorsen’s function. The third case to compare with Theodorsen’s function was for a combined pitching and plunging flat plate experiencing the same conditions and reduced frequencies as the two previous cases. The lift and wake shape results can be found in

Fig. 3-15. Similar conclusions can be drawn from Fig 3-15 as those in the pure plunging and pitching cases.

o o L(t) vs. U t/c, k = 0.1, h = +/- .05*c,  = +/- 5 z(x) vs. x, k = 0.1, h = +/- .05*c,  = +/- 5 o o o o o 0.5 0.5 L n=1 L n=10 0 0 L(t) L n=100 z(x) L Theodorsen -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x o o L(t) vs. U t/c, k = 0.5, h = +/- .05*c,  = +/- 5 z(x) vs. x, k = 0.5, h = +/- .05*c,  = +/- 5 o o o o o 0.5 1 L n=1 L 0.5 0 n=10 L(t)

L z(x) n=100 0 L -0.5 Theodorsen -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 U t/c o x o o L(t) vs. U t/c, k = 1.0, h = +/- .05*c,  = +/- 5 z(x) vs. x, k = 1.0, h = +/- .05*c,  = +/- 5 o o o o o L 0.5 0.5 n=1 L n=10 0 0 L(t)

L z(x) n=100 -0.5 L Theodorsen -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 -1 0 1 2 3 4 5 U t/c o x

Figure 3-15: Transient lift curves compared to Theodorsen’s function and wake shapes for combined pitching and plunging flat plate at k = 1.0, 0.5, and 0.1 show very good comparison between the DVM and Theodorsen’s function. With the implementation of unsteady aerodynamics into the code, validation was performed by comparing it with the Wagner function and Theodorsen’s function. The results against the Wagner function showed the dependency of time step size, as well as further proof of the importance of bound vortex number. With a high bound vortex number and a small time step size, the code did a good job approximating the Wagner function. The code was then used to compare with Theodorsen’s function and showed that it approximated the theoretical results well.

CHAPTER 4

HIGH ANGLE OF ATTACK CANONICAL CASES

4.1 High Angle of Attack Flow Field Validation

With the Wagner and Theodorsen functions successfully approximated in the previous section, the code was then used to make comparisons to experimental and CFD results for several AIAA Low Reynolds number Discussion Group canonical cases. The cases the DVM code was compared with were high angle of attack impulsively started airfoil, a linear ramp-hold, and a linear ramp up-hold-ramp down-hold.

The first validation case studied was airfoil impulsively started airfoil with α =

45o. The result was compared to experimental results gathered by Granlund, et. al.,2 in the AFRL Water Tunnel (HFWT). The flow parameters used in the analysis can be found in Table 4-1. Flow visualization comparison results were captured at various points throughout the evolution of the motion and can be seen overlaid in Fig 4-1.

Table 4-1: 45o impulsive start flow parameters.

3 V∞ (m/s) c (m) ρ∞ (kg/m ) μ∞ (Pa-s) Re

0.1524 0.1524 1000 1.00E-03 23.0E+04

α z(x) n Δt (s) t (s)

45o Flat Plate 20 1/30 7

Excellent comparison between the experimental and numerical results was obtained. Both show a large trailing edge and leading edge vortex (which was close to the airfoil surface). In every image, the DVM results come off at the same angle as the experimental results. The DVM LEV also seems to be about the same size as the experimental results at every instance in time. One interesting result was that the DVM

LEV particles did not enter the dark recirculation region near the trailing edge until the

LEV began to detach and roll back onto the airfoil surface. The particles seemed to stay on the border of the recirculation region before rolling back. In addition, at the times of

4.633 and 4.967 seconds, the experimental results show the leading edge vortex rolling back onto the airfoil while the numerical results only show the LEV detaching from the vortex convecting into the wake. This could be a result of water tunnel solid blockage or wall interactions, viscous effects not modeled, or spanwise effects.

Figure 4-1: Experimental results performed by Granlund, et. al., 2 compared to the numerical results of the leading edge and trailing edge vortex wake shapes for an airfoil impulsively started with α = 45o showed a good comparison between the experimental and the results obtained by the discrete vortex code.

The second validation case analyzed was the AIAA FDTC-LRWG canonical 0o-

45o ramp case. The results were then compared to experimental flow visualization CFD results by Granlund, et. al.,2. The flow parameters used in the analysis can be found in

Table 4-2 while the ramping motion can be found in Fig. 4-22.

Table 4-2: 0o-45o canonical case flow parameters.

3 V∞ (m/s) c (m) ρ∞ (kg/m ) μ∞ (Pa-s)

0.3048 0.1524 1000 1.00E-03

Re α z(x) n

46000 0o-45o Flat Plate 30

Ψ ζ DRF αsep

1.0 1.0 Planetary 16o

Figure 4-2: Angle of attack versus the physical time for the 0o-45o ramp-hold case presented by FDTC LRNDG2.

Figure 4-3 displays a flow field comparison of the code to experimental flow visualization results by Ol2 and CFD results by Garmann2 for nine angles of attack: 0.6o,

5.5o, 11.2o, 16.8o, 22.5o, 28.1o, 33.7o, 39.2o, 44.3o, and 45o. A number of interesting results can be seen in Figure 4-3. At the first angle of attack analyzed, both the PIV and

CFD results show what appear to be Helmholtz instabilities in the shear layer shed from the trailing edge while this does not occur in the DVM results. This occurred because both the bound and wake circulation strengths were zero when using the DVM until the airfoil pitched to a non-zero angle of attack. As the airfoil continued to pitch during the attached flow regime, the trailing edge vortex remained a steady line of circulation while the Helmholtz instabilities continued occurring in the PIV and CFD data.

Once separation occurred, the leading edge vortex modeled by the DVM compared well in size to the CFD results. However, it can be clearly seen that additional circulation was shed continuously from the airfoil surface in the CFD and PIV results while this did not occur in the DVM results. This occurred since vortices were only shed from the leading and trailing edge in the DVM. Though additional vortices could be shed from every location of the airfoil and thus potentially model this behavior, it would significantly increase the required computation time for a result that does not have a large impact on the flow field results at higher angles of attack.

When the angle of attack reached 22.5o, vortex instabilities occurred in the trailing edge wake. The occurrence of this instability compares very well with the instabilities that occurred in the PIV/Flow Visualization and CFD data, which were circled in the figure. The reason why these instabilities occurred in the DVM will be explained later.

When the angle of attack reached 28.1o, each flow field result shows the trailing edge

71 vortex being pulling back onto the airfoil surface due to the interaction between the shear layer and the large LEV. There are some discrepancies between the CFD and DVM results. The CFD results show significant separation of the LEV from the airfoil surface due to the shear layer while the DVM does not show this because vortices from the trailing edge shear layer have not been pulled into the LEV yet. The DVM result at this angle of attack also shows the same extent of instability in the trailing edge shed shear layer as those shown by the experimental and CFD results.

When the angle of attack reached 33.5o, there was very good comparison between the DVM and the experimental/CFD results. Both show a well defined LEV convecting downstream while the trailing edge vortex (TEV) forms directly beneath the LEV. One interesting comparison is that the TEV in the experimental results convected further downstream than the TEV in the numerical results (which agree very well) and was likely a result of blockage in the water tunnel. There is disagreement between the DVM and the other results when the angle of attack was 39.2o. In the DVM results, the TEV was a large vortex still relatively close to the trailing edge. However, the CFD results show a completely separated vortex with the shear layer preparing to be pulled back onto the airfoil surface while the experimental results show the TEV convected further downstream (again likely caused by blockage).

When the angle of attack reached 44.1o, there was slight disagreement between the DVM and other results. In both the experimental and DVM results, the TEV seemed to be reattaching to the airfoil while the TEV appeared to only have begun the process of reattachment in the CFD results. However, all of the results show a similarly sized and structured LEV. When the angle of attack reached 45o, there was a good similarity

72 between the DVM and other results. The numerical results show a well defined TEV beneath the LEV, though the CFD results show a more separated LEV than the DVM results. The experimental results also show the TEV to have convected farther downstream than in the numerical results (again due to blockage). All of the results show the shear layer from the trailing edge interacting with the large LEV (modeled in the

DVM as vortices being pulled into the LEV).

One interesting result was that the code modeled instabilities in the “shear layer” created by the finite vortices. The reason for this will be described later. However, the instances of the instabilities occurring can be seen in the figure and are circled. By visually analyzing the location of the instabilities in the figure, it can be seen that the shear layer instabilities occur in approximately the same location and to the same degree.

There were several factors that would have affected how well the DVM method matched the experimental and CFD results. The first was that the DVM is a low order, algebraic method in which only two vortices are shed (LE and TE) at each time step instead of a high order, differential method in which tens of thousands of interconnected grid points affect vortex interaction. Another cause is that viscous effects were only modeled by giving the point vortices a finite radius as opposed to full viscous effects (or a very high approximation) occurring in both the experimental and CFD results. Also, the DVM implemented modeling the airfoil used as a flat plate instead of incorporating the thickness effects of the SD7003 airfoil (which was used in the experiments and the

CFD) and thus would also alter the vortex interaction.

Although the DVM results compare very well to the CFD and experimental results, there was still an opportunity to make the DVM match better in terms of scale

73 and convection distance of the wake vortices. In order to do this, the DVM would have to be tuned to provide an optimal solution both in data comparison and computation time.

In the code, there were several parameters that, when adjusted, provided different but comparable results. Therefore, several sensitivity studies were performed in an attempt to tune the model so that the best results could be obtained. By finding a good value for various parameters that compare better with the experimental and CFD results, more confidence could be put in future uses of the code where alterations to the canonical case are made (such as varying the pitching function).

Figure 4-3: Experimental (left), low order model (middle), and CFD (right) vorticity and flowfield results for the 45o ramp-hold from 0.6o to 45o show a good comparison2.

The third validation canonical case analyzed was the AIAA FDTC-LRWG canonical 0o-40o-0o ramp up-hold-ramp down case. The results were then compared to experimental flow visualization and CFD results by Ol, et. al.,1. The flow parameters used in the analysis can be found in Table 4-3 while the ramping motion can be found in

Fig. 4-41.

Table 4-3: 0o-40o-0o canonical case flow parameters.

3 V∞ (m/s) c (m) ρ∞ (kg/m ) μ∞ (Pa-s)

0.1524 0.1524 1000 1.00E-03

Re Α z(x) n

23000 0o-40o-0o Flat Plate 40

Ψ ζ DRF αsep

1.0 1.0 Planetary 16o

 vs.  45

25  20

0 0 0.2 0.4 0.6 0.8 1 1.2 

Figure 4-4: Angle of attack versus the physical time for the 0o-40o-0o ramp-hold case presented1.

Figure 4-5 displays a flow field comparison of the code to experimental flow visualization results and CFD results by Ol, et. al.,1 for four angles of attack: 20o, 40o,

20o, 0o. As can be seen in the figure, the DVM code on the right matches the experimental and CFD flow field results very well. During the upstroke, a very small amount of shedding occurred at 20o due to the high speed of the pitching motion. As the airfoil continued pitching to 40o, the LEV only began to really form. During the downstroke, the LEV has formed and remains attached to the airfoil surface. In addition, a counter-rotating vortex was shed from the trailing edge due to the downward pitching motion. At the final angle of attack, the LEV stayed attached to the airfoil. In addition, the code models the counter-rotating vortex shed from the trailing edge due to the downstroke motion.

The experimental and CFD results both show the shear layer growing, which caused the LEV to separate from the airfoil at the end by the end of the downstroke. This however did not occur since the shear layer which caused the separation of the LEV was not modeled by the DVM. The other main difference between the code and higher order methods was that the distances between the TEV’s and the airfoil seem to be slightly different. However, even so, the code does a very good job approximating the higher order methods for this canonical case as well.

Figure 4-5: Experimental (left), CFD (middle two columns), and reduced order model (right) results for the 0o-40o-0o ramp-hold case shows good comparison between all four tests and for each angle of attack1. Several parametric studies were then performed to gage the sensitivity of the various parameters on flow field and vortex interaction to tune the model in order to obtain optimal results. This investigation will study the effects of:

 bound vortex number

 separated vortex strength factor

 separation angle of attack

 wake vortex radius size factor

 desingularization reduction factor

The first sensitivity study performed was on the effect that the number of vortices had on the flow field. Three angles of attack were analyzed for the sensitivity study

(~22.3o, ~37.5o, and 45o) for three different values of n (10, 30, and 60). The results can be found in Fig. 4-6, in which the left column corresponds to n = 10, the middle column corresponds to n = 30, and the right column corresponds to n = 60. A number of conclusions can be made. By comparing the flow fields in the first row in Fig. 7, it can be observed that as the number of vortices increases, the right edge of the LEV moves closer to the leading edge due to stronger vortices being shed from the leading edge since the free stream velocity is not convecting them as far. In addition, as the number of vortices increased, the flow field became more well-defined. The second and third rows of the figure also show a tighter LEV and TEV as the number of vortices increased. n = 10 n = 30 n = 60

Figure 4-6: Comparison of flow fields for three angles of attack by varying the number of bound vortices. The results show a more well-defined flow field as n was increased.

The second sensitivity study performed was on the effect that leading edge separated strength factor, Ψ, had on the flow field. The same three angles of attack as the previous sensitivity study were used to compare three different values of Ψ (0.7, 1.0, and

1.3). The results can be found in Fig. 4-5, in which the left column corresponds to Ψ =

0.7, the middle column corresponds to Ψ = 1.0, and the right column corresponds to Ψ =

1.3. A number of conclusions can be made. By comparing the flow fields in the first row in Fig. 4-7, it can be observed that as the strength factor was increased, the flow field experienced an interesting effect. With a lower strength factor, vortex instabilities can be seen in the leading edge vortex located at approximately -0.025 m. As the strength factor increased, the vortex instabilities disappeared. This occurred because the time step size was dependent on the strength factor. For a lower strength factor, a smaller time step size was used. This allowed for more vortices to be shed into the wake and thus cause the vortex instabilities. However, at the higher angles of attack, the angle at which the flow separates from the leading edge changes as the strength factor is increased. In addition, the wake vortex interaction changed as well. It should also be mentioned that since the strength factor and the bound vortex number are intimately related through the segment length size, there would be cross-coupling effects in the results.

Ψ = 0.7 Ψ = 1.0 Ψ = 1.3

The third sensitivity study performed was on the effect that the flow separation angle of attack had on the flow field. The same three angles of attack as the previous

o o o sensitivity studies were used to compare three different values of αsep (12 , 14 , and 16 ).

o The results can be found in Fig. 4-8, in which the left column corresponds to αsep = 12 ,

o the middle column corresponds to αsep = 14 , and the right column corresponds to αsep =

16o. A number of conclusions can be made. As the separation angle of attack increased, the size of the LEV seemed to decrease. In addition, at higher angles of attack, the trailing edge vortex seems to get closer to the airfoil as the separation angle of attack increased. This behavior is expected since the vortices have had additional time to convect downstream due to the lower angle of attack at which the flow separated.

o o o αsep= 12 αsep= 14 αsep= 16

Figure 4-8: Comparison of flow fields for three angles of attack by varying the flow separation angle of attack.

The fourth sensitivity study performed was on the effect that the size of the shed vortex radius compared to the size of the bound vortex radius. The same three angles of attack as the previous sensitivity studies were used to compare three different values of δ

(0.5, 1.0, and 1.5). The results can be found in Fig. 4-9, in which the left column corresponds to δ = 0.5, the middle column corresponds to δ = 1.0, and the right column corresponds to δ = 1.5. It can be seen when comparing the results for each angle of attack, changing vortex radius in the range of 0.5 to 1.5 did not have any significant effect on the flow field. δ= 0.5 δ= 1.0 δ= 1.5

Figure 4-9: Comparison of flow fields for three angles of attack by varying the shed vortex radius. The results show that varying the shed vortex radius did not have a significant impact on the vortex interaction.

The fifth sensitivity study performed was on the effect that the desingularization reduction function had on the vortex interaction. For this study, only one angle of attack was analyzed: 30o. The results can be found in Fig. 4-10. One result that can be seen in the figure is that the Low Order Algebraic, Gaussian, and Super Gaussian desingularization reduction functions do not provide significant vortex feeding shear layer instability in the LEV near the leading edge. The Planetary, High Order Algebraic, and Super Algebraic on the other hand do result in a strong level of instability in the LEV feeding shear layer.

The Low Order Algebraic and Gaussian did not have any shear layer vortices at the current angle of attack while the others did. Since the vortex radius is directly related to bound vortex number, there would be a cross-coupling effect between the bound vortex number and the desingularization reduction function due to the varying size of the vortices as the bound vortex number changes. This would then extend to the wake vortices since their radius is dependent on the bound vortex radius.

With leading edge separation modeled in the DVM, flow field comparisons were done between the code and experimental and CFD results for various AIAA canonical cases. The results obtained showed a good approximation of the higher order methods for each case analyzed. The location, size, and shedding frequency of the LEV and TEV matched well. Parametric studies also showed that the most important tuning parameter for flow field purposes was the bound vortex number, while the separation strength factor and desingularization function settings also had a decent influence.

4.2 The Uhlman Method

Once flow field validation was performed on the high angle of attack pitching maneuvers, specifically the 0o-45o ramp hold canonical case, it was necessary to calculate the forces and their respective coefficients on the airfoil. This task was initially approached using the Unsteady Bernoulli Method (UBM). As can be seen in Fig. 4-11, the method does a good job predicting the lift coefficient until just past a convective time of 4. Once flow separation occurred at the leading edge, the UBM failed due to the additional wake terms shed from the leading edge which did not incorporate surface effects due to the wake vorticity.

C vs.  L 2

-1 L C -2

-3

-4

-5 0 2 4 6 8 10 12 

Figure 4-11: CL vs. convective time using the UBM shows good lift coefficient prediction until leading edge separation occurred.

In order to take this new wake term into account and yield a good prediction of lift and drag coefficients, a different method had to be employed, which in this case was the Uhlman method. The Uhlman method24 for two dimensional flow (Eq. 4-124) is a method based on the total enthalpy, H, of the system, incorporates vortex motion through the volume integral, and temporal and viscous effects through the surface integral. In the equation, the G term is Green’s function represented by Eq. 4-224 for two dimensional flow while β has a value of π on a surface and 2π in a volume. Once the Uhlman method calculates the total enthalpy (Eq. 4-324), the Bernoulli equation is used to calculate pressure (Eq. 4-424). Once the pressure is found, the lift and drag (as well as their respective coefficients) can be easily calculated.

∞ (4-1)

∞

(4-2)

(4-3)

∞ ∞

(4-4)

∞ ∞

The implementation of the Uhlman method first required the generation of points along the upper and lower surface of the airfoil as shown as the blue points in Fig. 4-12

(where the magenta arrows are the surface points’ normal vectors) since pressure had to

o be calculated at the airfoil surface. -3 z(x) & ,  = 45.00 x 10 4

0.025

3 0.02

2 0.015

1 0.01

0 z(x) 0.005

-1 0

-2 -0.005

-3 -0.01

-4 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Figure 4-12: Points (blue) were created on thex upper and lower surface of the airfoil so that the Uhlman method could be used with the DVM

The next step involved generating the “influence” matrix of the total enthalpy, which incorporated the β factor and the surface integral of the change of G with respect to the normal vector (Eq. 4-524). A matrix was then constructed, as shown in Eq. 4-624, which would be inverted to find the total enthalpy at each surface point.

(4-8)

∞ ∞

(4-5)

π π

π (4-6)

π The next step was calculating the volume integral which incorporated the vortex motion, described by Eq. 4-724.

∞ κΓ ∞

(4-7)

κΓ ∞

Once the volume integral is found, the total velocity induced by all bound and wake vortices at each surface point first had to be calculated for the surface integral terms of the equation. With the velocity found at each surface point, the two surface integrals were calculated using eq. 4-824 and 4-924, which incorporate temporal and pseudo-viscous effects. The summation of the volume and surface integrals becomes the Right Hand

Side term (RHS). The total enthalpy at each surface point can then be calculated using

Eq. 4-1024. Once the total enthalpy is known, the pressure at each surface point is calculated using Eq. 4-1124. By knowing the pressure at each surface point, the lift and drag can be calculated using equations 4-12 and 4-13. With the lift and drag known, the lift and drag coefficients can be calculated.

(4-9)

π (4-10)

(4-11)

∞ ∞

(4-12)

(4-13)

Once the pressure is found at each surface point, the same procedure can be applied to points in the flow field so that pressure contours can be generated. However, when applying the Uhlman method to a flow field, the β must be changed from π to 2π since the analysis has moved to a volume as opposed to a surface.

4.3 Pressure, Force, and Force Coefficient Calculations Using the Uhlman Method

Once the Uhlman method was successfully implemented into the DVM code, it was used to calculate the lift and drag coefficients for the ramp-hold canonical case.

Figure 4-13 displays the lift and drag coefficients versus convective time. A number of things can be concluded by the results. The result showed that the Uhlman method does a credible job predicting the lift coefficient during the entire range of convective time for

α = 45o. The results of the Uhlman method post-flow separation showed the slope of the lift coefficient increasing due to the growth of the LEV. Then, there was sinusoidal variation in lift coefficient due to von Karman vortex shedding. This increase in lift coefficient is due to the new vortex terms shed from the leading edge causing additional vortex lift.

 vs. , k = 0.03 60

(deg) 20  0 0 5 10 15 20 25 30 35 40 45 convective time,  C vs.  L 3

2 L

C 1

0 0 5 10 15 20 25 30 35 40 45 convective time,  C vs.  D 3

2 D

C 1

0 0 5 10 15 20 25 30 35 40 45 convective time, 

One aspect of the coefficient results was that the peaks during the von Karman shedding were not sinusoidally shaped like the initial stall peak, but had an initial peak, dip, then a second peak until the TEV grew large enough to cause the drop in lift and drag coefficient. This was most likely caused by shear layer vortices flowing up the airfoil top surface, causing changes in velocity and thus pressure. Therefore, it can be determined that there was a Strouhal relationship with the shedding frequency if the middle of the two peaks is considered the central peak. It can also be deduced that the vortex shedding did experience periodicity.

To quantify this periodicity, the convective shedding period was taken between each peak in Fig. 4-13 and put in Table 4-4 and 4-5, where the first uses the chord as the physical length and the second uses the projected chord. The shedding period was then related to the convective shedding period using eq. (4-14) and through algebraic manipulation, solved for the shedding period in equation (4-15).

(4-14)

(4-15)

Since the free stream velocity was 12 inches/second while the chord was 6 inches, the convective shedding period was divided by two. By using equation (4-16), the shedding frequency was calculated.

(4-16)

The convective times were then subtracted to find the convective shedding period.

The convective shedding period was then divided by two and inverted to yield the shedding frequency. The average shedding frequency was found to be 0.482 Hz,

93 resulting in an average Strouhal number of 0.24 and 0.17. Comparing it with the Strouhal number for a cylinder as a theoretical bluff body comparison (0.21 at a Reynolds number of 1,000+), an average percentage difference of 13.55% and -19.71% was obtained.

Table 4-4: Shedding frequency results for 0o-45o ramp hold case where the physical dimension was the chord.

Peak (s) f (Hz) St StCyl Percent Difference 1 6.5 2 11 4.5 2.25 0.444444 0.219913 0.21 4.72% 3 15.25 4.25 2.125 0.470588 0.23285 0.21 10.88% 4 19.25 4 2 0.5 0.247403 0.21 17.81% 5 23.4 4.15 2.075 0.481928 0.23846 0.21 13.55% 6 27.3 3.9 1.95 0.512821 0.253746 0.21 20.83% 7 31.4 4.1 2.05 0.487805 0.241368 0.21 14.94%

Avg 4.15 2.075 0.481928 0.238957 0.21 13.79 %

Table 4-5: Shedding frequency results for 0o-45o ramp hold case where the physical dimension was the projected chord.

Peak (s) f (Hz) St StCyl Percent Difference 1 6.5 2 11 4.5 2.25 0.444444 0.075215 0.21 -64.18% 3 15.25 4.25 2.125 0.470588 0.149673 0.21 -28.73% 4 19.25 4 2 0.5 0.17494 0.21 -16.70% 5 23.4 4.15 2.075 0.481928 0.168617 0.21 -19.71% 6 27.3 3.9 1.95 0.512821 0.179426 0.21 -14.56% 7 31.4 4.1 2.05 0.487805 0.170673 0.21 -18.73%

Avg 4.15 2.075 0.482931 0.153091 0.21 -27.10 %

Comparing the lift and drag coefficients versus angle of attack data in Fig. 4-14 and Fig. 4-14 with data gathered by Ol2, et. al., shows a good comparison between the

DVM results and CFD/Water tunnel results. Figure 4-14 shows the lift coefficient reaching a maximum between 2 and 2.6 at stall, then von Karman shedding causing a lift coefficient mean of approximately 1.6 for all three cases. The drag coefficient data in

Fig. 4-15 shows von Karman shedding and a maximum drag coefficient between 1.5 and

1.8. The shedding frequency between the DVM and other methods were shown to be consistent with each other.

C vs.  L 3

DVM 2.5 SD7003, 50K SD7003, 20K 0.2 Span CFD 2 FPT

1.5 L C 1

0.5

-0.5 0 5 10 15 20 25 30 35 40 45 

o o Figure 4-14: CL vs. angle of attack plot for the 0 -45 ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers and CFD data for a 0.2 span SD7003. Results show a good comparison, though the max lift coefficient using the DVM is higher than the results for the SD7003 at a Re of 50K2.

C vs.  D 3

2.5 DVM SD7003, Re = 50K SD7003, Re = 20K 2 0.2 Span CFD FPT 1.5 D C 1

0.5

-0.5 0 5 10 15 20 25 30 35 40 45 

o o Figure 4-15: CD vs. angle of attack plot for the 0 -45 ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers. Results show a good comparison, though the peaks and troughs during the von Karman shedding do differ between using the DVM is higher than the experimental results for the SD70032.

Further comparison between the lift coefficient produced by the DVM and other methods was performed. In Fig. 4-164, the DVM results were plotted against various

CFD results, time-delay model analytical results, and preliminary water tunnel results.

As can be seen in the figure, the DVM does a good job in comparison with the other methods. The stall lift coefficient seemed to be the same between the DVM and CFD results. However, there seemed to be some differences in temporal location of when von

Karman shedding occurred. This occurred because the DVM results show an increase in lift coefficient slope once the flow separated at the leading edge, creating a low pressure region. The increased slope offset the DVM stall lift coefficient with respect to convective time, as it did in the angle of attack comparisons. This offset shifted the von

Karman shedding with respect to convective time, producing the observed difference

96 between when von Karman shedding occurred in comparing the CFD and DVM results.

An even closer inspection with respect to the time delay model in Fig. 4-17 shows that the code does a good job approximating the second analytical model while it does not make a good approximation on the first one. This was due to the mixing parameters of the time delay model used for the two different methods.

It was interesting that the increased slope did not occur in the CFD results. The cause for this result was that the pressure calculation was simply determined based on vortex lift, as stated earlier. As vortices were shed from the leading edge, they imparted a velocity vector in the same direction to the velocity vector induced by the bound vortex, which increased the velocity at the surface point. Since the velocity at the surface point was increased due to this effect, there would then be a decrease in pressure at this location, and thus yield an increase in the lift coefficient’s slope. Since pressure calculated by the CFD incorporated additional higher order terms (such as viscosity), there would be a difference. In addition, there would be an effect since CFD used velocity calculated in cells and not at discrete points using discrete vortices. Even so, it can be determined that the implementation of the separation strength function as the LEV shedding mechanism yielded good results both with respect to the flow field and generating lift coefficient data at an angle of attack of at least 45o.

Comparison of Aero Methods on 45 Ramp and Hold 3.5

CFD - Overflow 3 CFD - FL3DI - 2D CFD - FL3DI - 3D Analytic #1 2.5 Analytic #2 Water Tunnel DVM 2 L C 1.5

0.5

0 -5 0 5 10 15 20 Convective Time,  = t U / c o

Comparison of Aero Methods on 45 Ramp and Hold 3.5 Analytic #1 Analytic #2 3 DVM

2.5

2 L C 1.5

0.5

0 -5 0 5 10 15 20 Convective Time,  = t U / c o

Pressure, velocity, and vorticity contours were then constructed for four angles of attack (18.75o, 26.18o, 33.70o, 41.21o) and can be found in Fig. 4-18-4-21. As can be seen in the figure, the LEV creates a low pressure region above the airfoil, which induces an increase in the lift coefficient’s slope. As the LEV continued growing, so did the size of the low pressure region. The velocity contours show shear layers on the outer edge of the LEV. As the TEV rolled up onto the airfoil, it also created a low pressure region. It was the TEV rollup which caused the drop in lift post-stall. The streamline plots also show that the LEV adds shape to the airfoil since the streamlines flow around the LEV instead of passing into it.

Figure 4-18: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 18.75o.

Figure 4-19: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 26.18o.

100

Figure 4-20: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 33.7o.

Figure 4-21: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 41.21o.

After using the DVM to calculate the lift coefficient on the 0o-45o ramp hold case, calculations were performed on the 0o-85o ramp hold case using the same reduced frequency of 0.0324. The lift and drag coefficient results can be found in Fig. 4-22. The

101 plot shows that during the pitch up maneuver, the airfoil stalled at its maximum lift coefficient, then von Karman shedding occurred. However, as the airfoil pitched to higher angles of attack, the von Karman peak decreased since the vertical component of the normal force induced by pressure became less. The lift coefficient continued to decrease until it reached approximately 0.3.

The drag coefficient also increased until stall, slightly decreased due to von

Karman shedding, then increased to approximately 5 due to the low pressure regions created by the LEV and the horizontal component of the normal vector being significant.

The drag coefficient was extremely high since experiments19 have shown that the drag coefficient for a flat plate perpendicular to the flow is 2. This was most likely caused by the pressure difference (which is low due to the LEV and TEV) acting primarily in the horizontal direction or using too much resolution on the flow field. For similar reasons as stated in the 45o case, there did appear to be a Strouhal number related to the vortex shedding if the shedding period was taken between the middle of each twin peak.

102

 vs. ,  = 85o & k = 0.03 Max 100

50 (deg)  0 0 5 10 15 20 25 30 35 40 45 convective time,  C vs.  L 3

2 L

C 1

0 0 5 10 15 20 25 30 35 40 45 convective time,  C vs.  D 6

4 D

C 2

0 0 5 10 15 20 25 30 35 40 45 convective time, 

To find the shedding frequency, the same method used earlier was employed.

Table 4-6 and 4-7 displays the results, where the first uses the chord as the physical length and the second uses the projected chord. The average shedding frequency was around 0.52, yielding an average Strouhal number of 0.26 for both physical lengths.

Comparing it with the Strouhal number for a cylinder (0.21 at a Reynolds number of

1,000+), a percentage difference of 24.01% and 23.54% was obtained.

103

Table 4-6: Shedding frequency results for 0o-85o ramp hold case where the physical dimension was the chord.

Peak (s) f (Hz) St StCyl Percent Difference 1 6.7 2 11.5 4.8 2.4 0.416667 0.206169 0.21 -1.82% 3 15.6 4.1 2.05 0.487805 0.241368 0.21 14.94% 4 19.8 4.2 2.1 0.47619 0.235622 0.21 12.20% 5 21 4.3 2.15 0.465116 0.230142 0.21 9.59% 6 25.5 4.5 2.25 0.444444 0.219913 0.21 4.72% 7 29.5 4 2 0.5 0.247403 0.21 17.81% 8 35 5.5 2.75 0.363636 0.179929 0.21 -14.32%

Avg 4.32 2.16 0.46332 0.228318 0.21 8.75 %

Table 4-7: Shedding frequency results for 0o-85o ramp hold case where the physical dimension was the projected chord.

Peak (s) f (Hz) St StCyl Percent Difference 1 6.7 2 11.5 4.8 2.4 0.416667 0.070514 0.21 -66.42% 3 15.6 4.1 2.05 0.487805 0.155149 0.21 -26.12% 4 19.8 4.2 2.1 0.47619 0.204054 0.21 -2.83% 5 21 4.3 2.15 0.465116 0.216236 0.21 2.98% 6 25.5 4.5 2.25 0.444444 0.219077 0.21 4.32% 7 29.5 4 2 0.5 0.246461 0.21 17.36% 8 35 5.5 2.75 0.363636 0.179244 0.21 -14.65%

Avg 4.49 2.24 0.450551 0.184395 0.21 -12.19 %

Comparing the lift and drag coefficients for both the 0o-45o and 0o-85o in 4-23 showed the same lift and drag coefficients until the 0o-45o pitched motion ended at its maximum angle of attack. As the airfoil pitched past 45o, the lift coefficient dropped in comparison to the 0o-45o case while the drag increased well past the lift coefficient since the horizontal component experienced more of the normal force than the vertical component and as the airfoil pitched to high angles of attack, the LEV shed created larger pressure drops. Figures 4-24 and 4-25 plotted the lift and drag coefficients versus angle

104 of attack. According to the figure, after the angle of attack passed 45o, the drag coefficient overtook the lift coefficient since the airfoil’s horizontal component of the pressure induced normal force was greater than the vertical component. However, for reasons previously stated, the magnitude of the maximum drag coefficient seemed too high.

Comparing the code with the experimental and CFD results show a very good comparison. The shedding frequency and peaks seemed to match well while the various methods also over predicted the flat plate theory results. It is interesting that the overpredicted drag coefficient approximated the 20% span CFD results while the stall lift coefficient for both the DVM and CFD matched, though the second lift coefficient peak greatly over approximated the results.

 vs. , k = 0.03 100

50 (deg)  0 0 5 10 15 20 25 30 35 40 45 convective time,  C vs.  L 3

2 L

C 1

0 0 5 10 15 20 25 30 35 40 45 convective time,  C vs.  D 6

4 D

C 2

0 0 5 10 15 20 25 30 35 40 45 convective time, 

105

C vs.  L 3 DVM SD7003, 50K 2.5 SD7003, 20K 0.2 Span CFD FPT 2

1.5 L C 1

0.5

-0.5 0 10 20 30 40 50 60 70 80 

o o Figure 4-24: CL vs. angle of attack plot for the 0 -85 ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers and CFD data for a 0.2 span SD7003. Results show a good comparison, though there is more vortex shedding in the code than in the experimental data while it make a good approximation to the CFD data2.

106

C vs.  D 6

5 DVM SD7003, Re = 50K SD7003, Re = 20K 4 0.2 Span CFD FPT

3 D C 2

-1 0 10 20 30 40 50 60 70 80 

o o Figure 4-25: CD vs. angle of attack plot for the 0 -85 ramp hold case using the Uhlman method with comparison to experimental data on an SD7003 airfoil at two Reynolds numbers. Results are good up to 45o, than the DVM code significantly over predicts the experimental data while it does a good job predicting the CFD data2.

Pressure, velocity, and vorticity flow fields were then plotted at higher angles of attack in Fig. 4-26-4-28: 67o, 75o, and 85o. The low pressure regions can clearly be seen in the pressure contour while the shear layers are visible in the velocity contour. The von

Karman shedding can clearly be seen in the contour plots as well. In the left top corner image in all three figures, the streamlines wrap around the LEV. This shows the effect of the LEV in which it adds shape to an airfoil. The convection of the low pressure regions, which create the sinusoidal shape of the lift and drag coefficient during von Karman shedding, are visible in the three figures.

107

Figure 4-26: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 67.0o.

Figure 4-27: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 75.0o.

108

Figure 4-28: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 85.0o.

The final case that was analyzed using the Uhlman method was revisiting the 0o-

40o-0o pitch up-hold-return canonical case. The lift coefficient results can be found in

Fig. 4-29. As seen in previous cases, the Uhlman method did a good job calculating the lift coefficient on the pitch up. As seen in the figure, the lift coefficient on the pitch up started very high due to the high reduced frequency of 0.7. The lift coefficient slope increased once separation occurred due to the growth of the LEV. However, it was of interest to determine how well the method calculated lift coefficient on the down stroke.

Once the pitching motion finished, there was a drop in lift coefficient. This drop in lift coefficient continued during the down stroke motion, which eventually became negative due to the high speed downward pitching motion, as well as the upwash created by the counter-rotating vortex shed from the trailing edge. When the airfoil reached 0o, a counter-clockwise rotating vortex was shed from the trailing edge, which caused the jump in lift just past one convective time. Velocity, pressure, and vorticity contours were

109 then plotted in Fig. 4-30 - 4-33 for four angles of attack: 20o, 40o, 20o, 0o. The development of the LEV can clearly be seen in the figures. On the downstroke, the counter-rotating vortex shed from trailing edge (also mentioned in the previous section) was also evident.

 vs. 

20 (deg)  10

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 convective time,  C vs.  L 4

2 L C 0

-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 convective time, 

o o o Figure 4-29: CL vs. convective time for the 0 -40 -0 ramp hold case using the Uhlman method.

110

Figure 4-30: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 20 o.

Figure 4-31: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 40 o.

111

Figure 4-32: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 20 o.

Figure 4-33: Velocity, pressure, and vorticity contour plots for the pitching airfoil at an angle of attack of 0o.

112

Several parametric studies were then performed to gage the sensitivity of the various parameters on lift coefficient to tune the model. This investigation studied the effects of:

 importance of various terms on the RHS in the enthalpy/pressure

calculation

 bound vortex number

 separated vortex strength factor

 wake vortex radius size factor

 desingularization reduction factor

It was desirable to determine how important certain terms in the Uhlman method calculation were, i.e., the vortex motion volume integral, the temporal effects surface integral, and the viscous term surface integral. Therefore, a comparison was done for the ramp-hold case between three terms. This was done by ignoring the surface terms for one run, adding the surface effects into the second run, and adding the viscous terms into the third run, and adding all three terms into the fourth run. The resulting lift coefficient comparison can be seen in Fig. 4-34. It can be seen in the figure that the volume integral is the dominating term in the Uhlman method. It can further be seen that the viscous effects were in fact negligible due to the low kinematic viscosity of water.

113

C vs. ,  = 45o & k = 0.03 L Max 3 Volume Integral Only Temporal Surface Integral Added Viscous Surface Integral Added 2.5 All terms added

L 1.5 C

0.5

0 0 2 4 6 8 10 12 14 

The next study performed was to see the influence of bound vortex number.

Three values of n were selected (10, 30, and 60) for the ramp-hold case. As can be seen in Fig. 4-35, there was a very strong influence on the pressure and lift coefficient due to the bound vortex number. As the bound vortex number increased, the maximum lift coefficient increased. In addition, the convective times at which vortex shedding altered depending on the bound vortex number. As the vortex number and hence lift coefficient increased, the vortex shedding changed with respect to shedding frequency and duration.

The degree of dependence on bound vortex number was unexpected, though it does make sense. Since increased bound vortex number also increased the number of vortices in the wake, it was expected that a higher vortex number would create a large number of vortices in the wake, which would affect the stall lift coefficient and the shedding frequency. In addition, increasing the number of bound vortices should also

114 increase the accuracy of the solution. It should be noted that for this particular 6 inch chord flat plate, 30 vortices did a good job approximating the experiments and CFD.

This may have been a result of 30 vortices properly modeled the size of the physical boundary layer. However, for a differently sized plate/airfoil, 30 might over/under approximate the data. Therefore, when new cases are run using different sized plates or airfoils, a parametric study should be done to find out the right bound vortex number for that particular sized plate/airfoil, and then apply it to the new case

C vs. ,  = 45o & k = 0.03 L Max 3.5 n = 10 n = 30 3 n = 60

2.5

2 L C 1.5

0.5

0 0 2 4 6 8 10 12 14 

The next study performed was to see the influence of leading edge separation factor. Three values of Ψ were selected (0.7, 1.0, and 1.3) for the ramp-hold case. As can be seen in Fig. 4-36, there was a moderate influence on the pressure and lift coefficient due to the strength factor, which was directly tied to the number and size of the analysis’ time step. As the strength factor decreased, the maximum lift coefficient

115 increased slightly while the convective time of vortex shedding shifted as well since more vortices were shed into the wake. Due to its relationship with bound vortex number, the strength factor should be studied on a case by case basis and compare it with known data before applying it to new cases so that an appropriate strength factor can be used.

However, using a strength factor of 1.0 means that the vorticity shed from the leading edge is the same strength as the vorticity of the first vortex, which makes sense physically. Therefore, using a strength factor of 1.0 should be appropriate to most cases analyzed.

C vs. ,  = 45o & k = 0.03 L Max 3.5  = 0.7  = 1.0 3  = 1.3

2.5

2 L C 1.5

0.5

0 0 2 4 6 8 10 12 14 

The next study performed was to see the influence of the desingularization reduction function. As can be seen in Fig. 4-37, there was a strong influence on both the slope of the lift coefficient and oscillation/noise that occurred. According to the figure,

116 the Low Order Algebraic and Gaussian reduction factors produced a very low slope, as well as a moderate amount of oscillation post leading edge separation. The High Order

Algebraic and Super Gaussian both produced a higher slope than the Low Order

Algebraic and Gaussian while a lower slope than the Planetary and Super Algebraic.

The Super Gaussian also produced a moderate amount of oscillation post leading edge separation. The best results were the Planetary and Super Algebraic functions.

They had the same slope pre-leading edge separation as the experiments, did not experience any oscillation post-separation, and maintained similar behavior during the entire convective time range. The reason why there was such a discrepancy, even in the attached flow region, was because outside one vortex radius, some of the functions still caused reduction in influence of the bound vortices. Since the surface points were placed one vortex radius from the plate’s surface, they would experience the reduced influence of their segment’s bound vortex, depending on the selected reduction function.

117

C vs. ,  = 45o & k = 0.03 L Max 3 Planetary Low Order Algebraic High Order Algebraic 2.5 Gaussian Super Algebraic Super Gaussian 2

L 1.5 C

0.5

0 0 2 4 6 8 10 12 14 

The final study performed was to see the influence of the wake vortex size factor.

As can be seen in Fig. 4-38, there was a no significant influence of the size factor on the lift coefficient. Although the stall lift coefficient kept the same value for each size factor, its location with respect to convective time did vary slightly depending on the size factor since the desingularization reduction function would be affected and thus cause the difference. However, the convective time of vortex shedding did not seem to be impacted very much with respect to the size factor.

118

C vs. ,  = 45o & k = 0.03 L Max 3.5  = 0.5  = 1.0 3  = 1.5

2.5

2 L C 1.5

0.5

0 0 2 4 6 8 10 12 14 

This section provided a different method that was implemented to calculate the pressure at various points on the surface of the flat plate. The results in this section showed very good behavior and comparison between the results using the DVM’s implementation of the Uhlman method and experimental/CFD results. Strouhal number calculations showed that for the 0-45o case fell between 0.16 and 0.24 while the 0-85o case fell around 0.22. The parametric studies on the various tuning parameters showed that by setting the bound vortex number to ~30-40, the leading edge separation strength factor to 1.0, the desingularization reduction function to the planetary model, and the wake size factor to 1.0 gave the best results when compared to experiments and CFD.

However, the drag coefficient over predicted the data once the plate passed an angle of attack of 45o, which could have been a result of too much resolution. It was also shown that the volume integral was the dominant term in the Uhlman method calculation of total

119 enthalpy. Contour plots for the various canonical cases showed the development of the low pressure region created by the growth of the leading edge vortex. The velocity contours showed behavior reminiscent of shear layers around the vortices shed from the

LE and TE.

4.4 Reduced Frequency Dependency

All previous studies on the canonical ramp hold case used a reduced frequency of k = 0.0324. However, it was of interest to determine how dependent the lift and drag coefficients were on the pitching rate, i.e., reduced frequency. To determine the reduced frequency dependency, the ramp hold case was run using four different reduced frequency values in the vicinity of the reduced frequency used in the canonical case:

0.025, 0.05, 0.075, and 0.1. The lift and drag coefficient results for the four reduced frequencies were plotted along with the pitching profiles in Fig. 3-39. Several things can be seen in this plot. As the reduced frequency increased, the slope of the lift coefficient with respect to convective time pre-flow separation increased as well. It can also be seen that as the reduced frequency increased, the magnitude of the stall lift coefficient increased. As the reduced frequency increased, the von Karman shedding occurred sooner since the airfoil arrived at higher angles of attack faster at the higher reduced frequencies. After the initial peak due to stall, the lift and drag coefficients were coincident due to the orientation of the normal vector.

120

 vs.  50 k = 0.025 k = 0.050 k = 0.075 (deg) k = 0.10  0 0 2 4 6 8 10 12 14 16 18 convective time,  C vs.  L 4

L 2 C

0 0 2 4 6 8 10 12 14 16 18 convective time,  C vs.  D 4

D 2 C

0 0 2 4 6 8 10 12 14 16 18 convective time, 

Figure 4-39: Comparison of lift and drag coefficients versus convective time for various reduced frequencies in the 0o-45o ramp hold case. It can be seen that the highest reduced frequency yielded the highest stall lift coefficient because the highest reduced frequency experienced the fastest change in pitch. To gage a better understanding of the reduced frequency dependence, the lift and drag coefficients were plotted against the angle of attack in Fig. 4-40 - 4-41. While the flow remained attached, the lift and drag coefficient slopes with respect to the angle of attack remained approximately the same for all reduced frequencies. However, when separation occurred, the highest reduced frequency results experienced the lowest slope increase. This result seemed a bit unexpected. It was expected that the lowest reduced frequency would result in the lowest slope due to the slow pitching rate. This was most likely due to the lowest reduced frequency giving more time for the LEV to develop, as well as creating a stronger LEV, which would influence the lift coefficient. On the other hand, as the reduced frequency increased, the LEV remained attached at higher angles of

121 attack, resulting in a both a higher stall lift coefficient and stall angle of attack. In addition, it can be seen that as the reduced frequency increased, there was a substantially greater increase in drag than increase in lift coefficient. Also in the figures, the lift and drag coefficient were significantly higher than the values predicted by Flat Plate

Theory19, which did not have a 2π slope.

C vs.  L 3

k = 0.025 2.5 k = 0.050 k = 0.075 k = 0.10 FPT 2

L 1.5 C

0.5

0 0 5 10 15 20 25 30 35 40 45 

122

C vs.  D 3

k = 0.025 k = 0.050 2.5 k = 0.075 k = 0.10 FPT 2

D 1.5 C

0.5

0 0 5 10 15 20 25 30 35 40 45 

Figure 4-41: Drag coefficient versus angle of attack for the four reduced frequency values. The highest reduced frequency yielded a substantially higher peak as compared to the lift coefficient data. Although the 0o-45o canonical case is of interest, for the purposes of MAV perching, a higher final angle of attack is required. Therefore, the same reduced frequency comparisons were performed on the linear 0o-85o ramp-hold case. The results can be found in Fig. 4-42. As seen in Fig. 4-42, the magnitude of the stall lift coefficient increased as the reduced frequency increased, which also occurred in the previous case.

It can also be seen that as the reduced frequency increased, the von Karman shedding phased out in the lift coefficient since the plate pitched too fast for the TEV to roll up and interact with the LEV. Post-stall, the lower reduced frequency cases show decreasing peaks until the motion finished due to von Karman shedding while the higher reduced frequency cases did not show this effect. This was because as the reduced frequency increased, the normal vector pointed more horizontally earlier and thus did not induce a significant effect on the vertical force. This was proven in the drag coefficient plot since

123 the von Karman shedding can still be seen. However, it is very likely that the drag coefficient result at the high angle of attack was non-physical.

 vs.  100 k = 0.025 k = 0.050 50 k = 0.075

(deg) k = 0.10  0 0 5 10 15 20 25 30 35 convective time,  C vs.  L 4

L 2 C

0 0 5 10 15 20 25 30 35 convective time,  C vs.  D 10

D 5 C

0 0 5 10 15 20 25 30 35 convective time, 

Figure 4-42: Comparison of lift and drag coefficients versus convective time for various reduced frequencies in the 0o-85o ramp hold case. It can be seen that the lowest reduced frequency yielded the highest stall lift coefficient because the low reduced frequency allowed for more time for the LEV to develop. Due to the correlation between the 0o-45o and 0o-85o data, the same scaled effect was expected in which the lowest reduced frequency had the greatest slope increase due to lower pitching rate allowing the LEV to have more time to develop. To show this, the lift and drag coefficients were plotted against angle of attack, as shown in Fig. 4-43 and

4-44. It can be seen that as the reduced frequency increased, the slope increase decreased as it did in the 0o-45o case. The figure also further shows that as the reduced frequency increased, the von Karman shedding phased out in the lift coefficient since the plate pitched too fast for the TEV to roll up and interact with the LEV until the pitching motion finished. All four reduced frequencies also ended at approximately the same lift coefficient when the pitching motion ended. It can be seen in Fig. 4-44 that as the

124 reduced frequency increased, the drag coefficient increased as well. As in previous case, the lift and drag coefficients were higher the Flat Plate Theory results.

C vs.  L 3 k = 0.025 k = 0.05 k = 0.075 2.5 k = 0.10 FPT

L 1.5 C

0.5

0 0 10 20 30 40 50 60 70 80 90 

Figure 4-43: Lift coefficient versus angle of attack for the four reduced frequency values. As reduced frequency increased, the von Karman shedding was eliminated due to the TEV not rolling up.

125

C vs.  D 9

k = 0.025 8 k = 0.050 k = 0.075 7 k = 0.10 FPT 6

5 D C 4

0 0 10 20 30 40 50 60 70 80 90  Figure 4-44: Drag coefficient versus angle of attack for the four reduced frequency values. As reduced frequency increased, the von Karman shedding was eliminated due to the TEV not rolling up. The final case analyzed was the 0o-40o-0o canonical case. Due to the higher reduced frequency of the canonical case, the reduced frequencies analyzed were: 0.25,

0.5, 0.75, and 1.0. The results can be found in Fig. 4-45. At lower reduced frequencies, the lift coefficient did not drop significantly during the down stroke since the LEV remained well attached. As the reduced frequency increased to higher values, the fast down stroke caused the lift coefficient to become negative even though the flow field plots show the LEV remained relatively well attached even at the reduced frequency of

0.7. The drag coefficient at the lower reduced frequencies decreased on the down stroke with the airfoil’s motion. At the higher reduced frequencies, the drag reached near zero before the pitching motion ended.

126

 vs.  k = 0.25 40 k = 0.50 k = 0.75 20 k = 1.0 (deg)  0 0 0.5 1 1.5 2 2.5 3 convective time,  C vs.  L 5

L 0 C

-5 0 0.5 1 1.5 2 2.5 3 convective time,  C vs.  D 5

D 0 C

-5 0 0.5 1 1.5 2 2.5 3 convective time, 

Figure 4-45: Convective time versus angle of attack, and lift and drag coefficients. The results obtained showed a strong dependency on reduced frequency post flow separation. As the reduced frequency was increased, the von Karman shedding was nearly eliminated during the pitching motion. The results also significantly over predicted flat plate theory due to both the transient motion and flow separation.

4.5 Application of DVM Code and Uhlman Method to Perching Maneuver

Now that the Uhlman method has been used to analyze the lift and drag coefficients on high angle of attack maneuvers, as well as their reduced frequency dependency, the code was used to calculate the lift and drag coefficients for a thin airfoil undergoing a perching-like maneuver. For this case, the angle of attack was held at 0o for five seconds, increased linearly with a reduced frequency of 0.16, and then held in a similar fashion as the canonical ramp-hold cases, as shown in Fig. 4-46.

127

 vs.  90

50 

0 0 10 20 30 40 50 60 70 

Figure 4-46: Angle of attack versus convective time profile for the “perching” case analyzed. The free stream velocity for the convective time calculation was taken to be twice the chord length. The most significant difference between the “perching” case and the previous canonical cases was that the free stream velocity in the horizontal direction was not constant. The horizontal velocity decreased (referred to as “surge” since the velocity decreased in the surge direction) with respect to time according to Fig. 4-47, where water tunnel data developed by colleagues that has not yet been published was used to create a curve fit (eq. (4-17)) which was used in the DVM code. Due to this, it did not seem suitable to plot time histories with respect to convective time. For this section, all time histories will be plotted against physical time.

(4-17)

128

Since the free stream velocity decreased during the motion, the dynamic pressure decreased as well, on the order of V2 at each time step as shown in Fig. 4-47. It can be seen by comparing the angle of attack and velocity in Fig. 4-48 that the airfoil flies horizontal for a period of time during which the velocity decreased, then the angle of attack changed while the velocity continued decreasing. Due to the very small time step and long computation time, the time step size was multiplied by three to decrease the computation time. Since the dynamic pressure decreased to zero, it was expected that the force coefficients would not be applicable when analyzing a perching-like maneuver.

Velocity vs. Time for True Perch in Water 0.35 True Perch Curve Fit 0.3

0.25

0.2

0.15 Velocity, m/s Velocity,

0.1

0.05

0 0 5 10 15 20 25 30 35 Time, s Figure 4-47: The velocity profile for the perching case using data from the water tunnel and a curve fit.

129

Velocity,  vs. Time 0.4 100

0.2 50 , deg ,  Velocity,m/s

0 0 0 5 10 15 20 25 30 35 t, s Dynamic Pressure vs. Time 50

2 30

20 Q, N/m Q,

0 0 5 10 15 20 25 30 35 t, s

Figure 4-48: The velocity, angle of attack, and dynamic pressure profile with respect to time during the perching case. The angle of attack and velocity profile plot shows that the airfoil flies horizontally while experiencing the decreasing velocity, then begins pitching up as the airfoil continued decreasing in velocity. The dynamic pressure decreased to zero which means that force coefficients will not be useful for a perching analysis. The lift and drag coefficients (based on instantaneous dynamic pressure) were then plotted against time in Fig. 4-49. As shown in Fig. 4-49 after the airfoil stalled, the lift continued increasing to 3.75 after the drop most likely due to the TEV rollup. After

30 seconds, there were spikes that occurred in the data. The reason for these two occurrences was due to the dynamic pressure decreasing. As the dynamic pressure approached zero, the lift experienced would approach a 0/0 type of behavior. Similarly, the drag coefficient increased to extremely high values for the same reason as stated above. As the dynamic pressure approached zero, the drag coefficient would experienced real number/0 and thus cause infinite drag coefficient. Therefore, for a perching maneuver in which the free stream decreases, it would be more beneficial to analyze the lift and drag force values as opposed to their accompanying coefficient values.

130

 vs.  100

50 (deg)  0 0 10 20 30 40 50 60 70 convective time,  C vs.  L 10

L 5 C

0 0 10 20 30 40 50 60 70 convective time,  C vs.  D 20

D 10 C

0 0 10 20 30 40 50 60 70 convective time,  Figure 4-49: Angle of attack, lift coefficient, and drag coefficient versus time. After the airfoil stalled, the lift and drag coefficient continued increasing since the dynamic pressure was getting small. Therefore, the lift force and drag force per unit span were then plotted against physical time in Fig. 4-50 and Fig. 4-51. According to Fig. 4-50, the lift was zero until the angle of attack began increasing. Due to the decreasing free stream velocity, the lift in the attached flow region was not linear. Post-separation, the lift continued increasing to stall, at which the lift then dropped due to the TEV. The lift then increased slightly, decreased, increased slightly again due to von Karman shedding, then decreased to zero.

The drag also increased to stall, then dropped due to the TEV rollup, then increased to a plateau of 1.35, then decreased to near zero due to the velocity decreasing to zero.

131

L',  vs. Time 2.5 100

2 80

1.5 60 , deg ,  L, N/m L,

1 40

0.5 20

0 0 0 5 10 15 20 25 30 35 t, s

Figure 4-50: Lift per unit span versus time. After the airfoil stalled, the lift began dropping to zero while the von Karman shedding is causing some slight increases in lift.

D',  vs. Time 2 100

1 50 , deg ,  D, N/m D,

0 0 0 5 10 15 20 25 30 35 t, s Figure 4-51: Drag per unit span versus time. The drag force increased post-stall due to the high angle of attack but then dropped due to the decreasing free stream velocity. With respect to a bird or MAV perching, the lift and drag time histories can provide some insight. As the bird begins the climb phase of a typical perching trajectory and rapidly begins increasing in angle of attack, there is a large increase in lift due to the

132 pitching motion and creation of the LEV. Since the bird is moving vertically due to the lift and thus increasing in potentially energy, the bird’s velocity would decrease due to the loss of kinetic energy. As the LEV begins detaching during the climb phase, the bird would lose lift. As the wings continue pitching, the drag increases due to bluff body flow. The combination of losing lift, increasing drag, and the decreasing velocity due to the loss of kinetic energy would allow the bird to successfully perch.

Velocity, pressure, and vorticity contours were then plotted taken at various times during the maneuver: 5, 10, 15, 20, 25, 30, and 33 seconds (Fig. 4-52-4-58). As can be seen through the figures, the airfoil did not experience any vorticity and pressure difference until the angle of attack increased to a non-zero value. The LEV then began forming and growing in size. As the angle of attack continued increasing the TEV rolled up, but due to the decreasing velocity, did not experience the exact same TEV growth as in the canonical cases, though it was similar. As the airfoil reached 85o and the velocity continued decreasing, the convected TEV stayed relatively close to the airfoil as the LEV and attached TEV continued growing. The flow field at the final instant in time of the maneuver can be found in Fig. 4-59. Due to the decreasing velocity, the shed vortices did not convect as far from the airfoil while the most recently shed vortices did not have much strength compared to the vortices previously shed that had convected into the wake.

133

Figure 4-52: Velocity, pressure, and vorticity contours at 5 seconds. No vorticity has been developed, and thus no pressure difference to create lift or drag.

Figure 4-53: Velocity, pressure, and vorticity contours at 10 seconds. The LEV has begun forming and has created a low pressure region.

134

Figure 4-54: Velocity, pressure, and vorticity contours at 15 seconds. The LEV has begun separating from the airfoil surface due to the TEV and shear layer rollup.

Figure 4-55: Velocity, pressure, and vorticity contours at 20 seconds. The second LEV has begun forming while the TEV began separating.

135

Figure 4-56: Velocity, pressure, and vorticity contours at 25 seconds. The second LEV continued growing and thus intensifying the low pressure region above the airfoil surface.

Figure 4-57: Velocity, pressure, and vorticity contours at 30 seconds. The airfoil has arrived at 85o and the LEV has begun separating from the airfoil due to TEV and shear layer rollup.

136

137

o -3 z(x) & ,  = 85.00 x 10

0.4 1.5

0.3 1

0.2

0.5 0.1

0 0 z(x)

-0.1

-0.5 -0.2

-0.3 -1

-0.4

-1.5

-0.5 0 0.5 1 1.5 x

Figure 4-59: Full flow field plot during the perching maneuver. The vortices did not convect as far due to the decreasing velocity. A comparison between the growth of the LEV and TEV as the airfoil pitched up with both a constant free stream velocity and decreasing velocity at various angles of attack was then done. The results can be found in Fig. 4-60. As shown in the figure, the vortex interaction did not seem to vary significantly until the angle of attack reached high values. Though, it can be seen that the vortex strengths did vary. Comparing the flow field calculated by the DVM and the experimental results obtained by colleagues in Fig.

4-61 showed that the DVM did a good job approximating the experimental results. The shedding angle was the same for all four images, except at 45o. The DVM results at the lowest angle of attack showed that the vortices shed in the DVM code convected further past the LEV developed in the experiment’s flow visualization. At the higher angle of attack, the DVM again did a good job approximated the experimental results.

138

o -3 z(x) & ,  = 36.94 x 10 0.15 o5 -3 z(x) & ,  = 37.03 x 10 0.15 4 0.1 Constant V∞ Decreasing V∞ 1.5 o 3 α = 37 0.1 2 1 0.05

0.05 1 0.5

0 0 z(x) 0 0

z(x) -1

-0.05 -2 -0.5 -0.05 o o -3 z(x) & ,  = 64.55 -3 z(x) & ,  = 64.67 x 10 x 10 0.15 0.15 -3 2 5 -1 -0.1 -0.1 -4 4 -1.5 1.5 0.1 0.1 o -5 -0.15 α = 67 -0.15 3 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 1 x x 2 0.05 0.05 0.5 1

0 0 0 0 z(x) z(x)

-1 -0.5 -0.05 -0.05 -2

-1 -3 o -0.1 o -3 z(x) & ,  = 85.00 -0.1 x 10 -3 z(x) & ,  = 84.85 -4 x 10 0.15 -1.5 0.15 2 5 o -5 -0.15 -2 0 α = 850.05 0.1 0.15 0.2-0.15 0 0.05 0.1 0.15 0.2 4 x 1.5 0.1 0.1 x 3 1

2 0.05 0.05 0.5 1

0 0 0 0 z(x) z(x)

-1 -0.5 -0.05 -0.05 -2

-1 -3

-0.1 -0.1 -4 -1.5

-5 -0.15 -0.15 -2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.05 0.1 0.15 Figure 4-60: Flowx field comparison between thex decreasing free stream and constant free stream case shows that the flow fields did not vary significantly until the airfoil reached high angles of attack, when the velocity was near zero.

139

α = 25o α = 45o

α = 67o α = 85o

Figure 4-61: Flow field comparison between the DVM and experimental results obtained by Ol show a good comparison with some difference, especially at 45o.

140

With the canonical cases analyzed using the Uhlman method, a perching-like case was analyzed next. The velocity profile used for this perching case was a curve-fit to unpublished experimental results used by Michael Ol. Due to the decreasing velocity profile, and thus dynamic pressure, lift and drag coefficients were unreliable. Therefore, lift and drag per unit span force was used instead. The lift force showed a large peak, which then decreased to zero with some small bumps occurring. The contours showed the vortex interaction at various instances in time. A comparison of the flow fields at various angles of attack between the decreasing velocity profile and constant velocity profile showed some small differences. The results also showed a good comparison with experimental flow field images obtained by Michael Ol.

4.6 Vortex/Shear Layer Instabilities

According to Durst25, a shear layer is a region of fluid where the fluid’s velocity just above the layer is higher than the velocity just below the layer (or vice versa). Shear layers can be formed from jets, wakes, etc. In the DVM code, the shear layer is pseudo- modeled by wake vortices since they either increase or decrease velocity above/below them. By calculating the velocity in the flow field around the airfoil, a velocity contour was generated which allowed for plotting the pseudo-modeled shear layers. The trailing edge and leading edge shear layers can clearly be seen in Fig. 4-62 and Fig. 4-63, respectively.

141

Figure 4-62: Velocity contour where the trailing edge shear layer is visible when the airfoil is at an angle of attack of 11.23o.

Figure 4-63: Velocity contour where the trailing edge shear layer is visible when the airfoil is at an angle of attack of 22.46o. However, experiments and CFD have shown that instabilities can occur in the shear layer due to Kelvin-Helmholtz instabilities (as shown in Fig. 4-642) which cause a

142 roll-up in the fluid. Even though the code is significantly lower order than CFD and experiments, it was still of interest to see if the code modeled (or at the very least pseudo- modeled them).

Figure 4-64: PIV and CFD images showing the Kelvin-Helmholtz instabilities that occurred in the shear layer of the 0-45o ramp-hold canonical case.2

As described in section 4.1, vortex instabilities did result from executing the code.

As shown in figure 4-3, the instabilities propagated through both the leading edge and trailing edge wakes as they grew in size. An obvious task was to figure out why this occurred in such a low order method. To do this, it was necessary to plot the wake and velocity contours at various angles of attack during the ramp-hold pitching motion. The vortex instabilities in the trailing edge wake can be found in Fig. 4-65 while the instabilities in the leading edge vortex can be found in Fig. 4-66.

As can be seen in the top image of Fig. 4-65, a bump can be seen in the trailing edge wake. This was a result of the leading edge vortex being shed and causing a change in the bound circulation. With the Kelvin condition equation incorporating this new circulation term, the strength of the vortex shed from the trailing edge would have to be adjusted such that the total circulation remained zero. As the leading edge vortex

143 continued to grow, its influence on the bound circulation increased as can be seen in the subsequent images. As the bound circulation continued changing with respect to strength and sign due to the influence of the LEV, it caused a “rippling” effect in strength of the vortices shed from the trailing edge. With the strength of the trailing edge vortices constantly changing due to the growing influence of the LEV, the velocity imparted by the trailing edge vortices on each other then caused them to roll up into groups of vortices, which are similar to Kelvin-Helmholtz instabilities. However, in the images of the figure, it can be seen that instabilities had not begun propagating into the LEV as the instabilities in the trailing edge shear layer continue.

As can be seen in the top image of Fig. 4-66, the trailing edge shear layer separated and began to roll back onto the airfoil surface. As the TEV began growing, vortices from trailing edge shear layer were swept up the airfoil towards the LEV. A combination of the TEV growing and inducing its own “rippling effect”, as well as the vortices getting pulled into the LEV, was the cause of instabilities propagating in the

LEV. Therefore, it can be concluded that the strength and sign fluctuation of the bound circulation due to the influence of the LEV and TEV rollup were the cause behind the vortex/shear layer instabilities. From Fig. 4-66, it can be also deduced that there must some sort of periodicity involved in the instabilities and shedding frequency, as well as a spatial relationship.

144

o -3 z(x) & ,  = 18.75 x 10 0.15 4

3 0.1 2 0.05 1

0 0 z(x)

-1 -0.05

-2

-0.1 -3

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 x

o z(x) & ,  = 22.46 -3 x 10 0.15 4

3 0.1

0.05 1

0 0 z(x)

-1 -0.05 -2

-0.1 -3

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 x

o -3 z(x) & ,  = 26.18 x 10 0.15 4

3 0.1

0.05 1

0 0 z(x)

-1 -0.05

-2

-0.1 -3

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 x

Figure 4-65: Flow field and velocity contour showing the development of the trailing edge shear layer with the growth of the LEV.

145

o -3 z(x) & ,  = 28.12 x 10 0.15 4

3 0.1

0.05 1

0 0 z(x)

-1 -0.05

-2

-0.1 -3

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 x

o -3 z(x) & ,  = 29.98 x 10 0.15 4

3 0.1

0.05 1

0 0 z(x)

-1 -0.05

-2

-0.1 -3

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 x

o -3 z(x) & ,  = 33.69 x 10 0.15 4

3 0.1

0.05 1

0 0 z(x)

-1 -0.05

-2

-0.1 -3

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 x

Figure 4-66: Flow field and velocity contour showing the development of the leading edge shear layer with the growth of the TEV.

146

In addition to yielding a good approximation both in the flow field and the force/force coefficient calculations, the DVM code also modeled behavior reminiscent of shear layer instabilities. When leading edge separation occurred, it changed the rotation and strength of the bound vortices. As the LEV grew, the bound vortices kept changed which caused a rippling effect that induced strength changes in the TE vortices. This caused some vortices to be stronger than others, and effectively induce rotation around the stronger vortices. As the TEV began wrapping back onto the airfoil surface, this same rippling effect induced similar instability propagation in the LE vortices.

147

CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

The Discrete Vortex Method is a very powerful low order tool that has significant applicability in the design process of MAVs. The code developed was effectively used to analyze three classes of two dimensional problems: steady flow, low amplitude unsteady, and high alpha unsteady. As shown in Section 3.1, the circulation distributions for three geometries were compared with theoretical solutions and matched them with a high level of accuracy. It was during this set of analyses that the role of the bound vortex number began coming to light. In Section 3.3, the lift modeled by Wagner and Theodorsen’s functions was also approximated accurately by the DVM, thus proving the method’s effectiveness in low amplitude unsteady flow cases. The role of the time step size was also understood in this section. These two previous sections were mere validation cases for the code before looking at high alpha aerodynamics.

The most significant results were generated in the high angle of attack maneuvers, specifically through comparison with various canonical cases developed by the AIAA

FDTC-LRWG. Vortex shedding at the leading edge using the separation strength factor effectively modeled the creation of a LEV when compared with experimental and CFD results in Section 4.1. The Uhlman method was then used to calculate pressure at points along the airfoil’s upper and lower surface. This allowed for the calculation of the lift and drag coefficients of the airfoil at high angles of attack. Compared with experimental

148 and CFD results, the DVM provided an accurate low order approximation. The main difference between the DVM results was that once separation occurred, there was an increase in lift coefficient slope which did not occur in the higher order methods. This was a result of the pressure calculation being solely dependent on the velocity induced by the bound and wake vortices which caused a significant pressure drop and thus an increase in lift coefficient post-separation, as opposed to higher order methods which incorporated additional effects such as viscosity.

In addition, the lift and drag coefficient results once von Karman shedding occurred showed regions of a peak, then dip, then peak before the TEV rollup caused the drop in lift and drag coefficient as opposed to smoother sinusoidal shapes. This was most likely a result of the velocity induced by the shear layer vortices. However, if by taking the middle point in each dip as the “peak” during the von Karman shedding, a Strouhal relationship can be defined with respect to the vortex shedding frequency of the LEV and

TEV. It was then found that the shedding frequency was around 0.5.

At higher angles of attack (85o), the lift drop and drag increase were significant.

At high angles of attack, the vertical component of the pressure induced normal force on each airfoil segment becomes very small since the normal vector became near horizontal at the high angle of attack. In addition, the LEV at the high angles of attack created a very large pressure drop. Since the normal vector was near horizontal, the horizontal force (drag) became significant such that the drag coefficient was approximately 5.

The code showed in Section 4.3 that there was a reduced frequency dependency during the pitching motion. It was found that as the reduced frequency was increased, the stall lift coefficient increased as well. It was also shown that the lift coefficient slope

149 decreased and approximated flat plate theory as the reduced frequency was increased.

This was most likely a result of the lower pitch rate allowing for the LEV to grow more.

Studies were also done with regard to the various tuning parameters (bound vortex number, leading edge separation strength factor, wake vortex size factor, etc.).

The results showed that the optimal bound vortex number when compared to the experimental and CFD results were that 30 vortices created the most similar stall lift coefficient to the additional results. The leading edge separation strength factor set to 1.0 gave good results and yielded the most physically correct results. The planetary desingularization reduction function yielded the smoothest results and gave the best approximation to the data. The wake radius size factor of 1.0 gave the best results and also was the most physically appropriate. Finally, the comparison of terms in the Uhlman method total enthalpy calculation showed that the volume integral term was the most important term in all cases analyzed and would yield by itself good results.

The code also pseudo-modeled shear layer instabilities observed in experiments and high order computations in both the LEV and TEV, which was unexpected. This was most likely a result of a rippling effect induced by the LEV onto TEV rollup, which caused the bound vortices to constantly change strength and sign. As the TEV rolled onto the airfoil, vortices continued flowing into the LEV and induced counter-rotating velocities which caused instabilities to propagate into the LEV in addition to the same rippling effect that occurred in the TEV.

Although the code proved to be quite powerful when analyzing high angle of attack maneuvers, more work should still be done using it. One obvious extrapolation would be to extend the code from being two-dimensional to three-dimensional. Although

150 a significant amount of information can be obtained from two-dimensional analyses, a full investigation would require a three-dimensional component. This would incorporate wing tip vortices, which would interact with the vortices shed from the leading and trailing edges. The interaction would cause an effect on the forces that would not be taken into account with a simple two-dimensional analysis.

An additional step would be to move away from a flat plate by adding camber to the airfoil. This can be done in multiple ways. One way to add camber would be by adding a fixed amount of camber during the entire maneuver. This fixed camber would increase the stall lift coefficient and potentially cause the LEV to remain attached for a longer duration of time. Another way of adding camber would be using a variable camber approach. By starting with a certain amount of camber (whether that be zero or a small percentage of the chord), the camber would be increased during the pitching motion. This variable camber could also provide an effective means of controlling the

LEV with the additional of creating more lift due to the additional of more camber.

The code should also be applied to a full perching maneuver. The case studied in

Section 4.5 only incorporated the decreasing free stream velocity in the horizontal direction. A full perching maneuver would incorporate decreasing velocity in the horizontal motion and high angle of attack pitching, but also plunging style displacement since a perching trajectory would incorporate changes in altitude as well.

151

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155

APPENDICES

A-1 Obstacles Encountered

Since this implementation of the DVM involved coding in MATLAB, there would be a number of issues that would appear due to the nature of writing code. Most issues that were encountered involved a red pop-up in the MATLAB command window saying that MATLAB could not run. The pop-up would then tell where the issue is located and give an indication as to what the problem in the code might involve, as shown in Fig. A-1.

Figure A-1: Example of the red pop-up in MATLAB’s command window to indicate a problem in the coding. However, not all code problems illicit an immediate response from MATLAB indicating an issue. Some problems that were encountered in the writing of this code

156 involved improper variables but proper coding, which resulted in incorrect results. The first instance of this type of problem occurred after the code was updated to incorporate low angle of attack and low amplitude unsteady aerodynamics. When the code was finally ready to be used to compare with the Wagner function, the results were extremely perplexing, as shown in Fig. A-2. The problem was that the velocity components were incorrectly labeled. When the labeling was switched, the problem was solved. The correct solution can be found in Fig. A-3.

157

 vs. U t/c o 1.1  n=100, t=0.1 s  1 Approximation 1

0.9

0.8 

0.7

0.6

0.5

0 0.5 1 1.5 2 2.5 3 U t/c o Figure A-3: Corrected Wagner function plot using 100 bound vortices shows strong comparison between DVM and Wagner function approximation. The next set of problems occurred when leading edge separation was modeled into the code. Without making any adjustments, when vortices were shed from the leading edge, they could potentially pass through the airfoil (obviously incorrect), as shown by Fig. A-4. In order to correct this, a position vector was drawn from a point on the airfoil camberline to the vortex. A dot product was then taken between the position vector and the normal vector of the airfoil point. If the dot product was negative, then the vortex passed through the airfoil. To correct this, the vortex was moved by two times the horizontal and vertical component of the position vector, as shown in Fig. A-5. The corrected flow field in which the vortex remained above the airfoil can be seen in Fig. A-

6. A similar algorithm was used in case trailing edge vortices passed through the airfoil as well.

158

Figure A-4: Plot showing a vortex shed from the leading edge passing through the airfoil.

Figure A-5: Plot showing a how the vortex is moved from below the airfoil to above the airfoil.

159

Figure A-6: Plot showing corrected vortex shed from the leading edge without passing through airfoil. The most difficult problem to be solved involved the Uhlman method’s calculation of the lift coefficient during the ramp-hold canonical case. As can be seen in

Fig. A-7, as the airfoil pitched up, the Uhlman method calculated the lift coefficient very well in the attached region. When flow separation occurred, the Uhlman method captured the increased lift due to the low pressure region of the growing LEV to stall.

However, as the trailing edge shear layer rolled back onto the airfoil, non-physical spikes began showing up. Extending the convective time out, the anomalous behavior can be seen to a greater degree in Fig. A-8. Although numerous investigations were conducted to eliminate this issue, none of them proved fruitful. However, after months of re-reading the lines of code, the problem was solved.

One of the ways to eliminate some of the non-physical behavior was to prevent vortices from moving within a certain distance of the surface points. Implementing a similar method as that used to prevent vortices from cross the airfoil surface, LE and TE vortices were kept from flowing within 3 vortex radii of the airfoil. However, it took several months to see that the placement of one line of code caused the anomalous

160 behavior. By moving the line from the location in Fig. A-9 to A-10, the Uhlman method generated very good lift coefficient data post-stall as shown in Fig. A-11.

C vs.  o L -3 z(x) & ,  = 15.00 x 10 1.4 0.15 4 1.2 0.1 3

2 1 0.05 1 0.8

0 0 C

z(x) 0.6 -1 -0.05 0.4 -2 -0.1 -3 0.2

-4 -0.15 0 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5  x o -3 z(x) & ,  = 18.75 x 10 0.15 4 C vs.  L 1.8 3 0.1 1.6 2 1.4 0.05 1 1.2

0 0

z(x) 1 L C 0.8 -1 -0.05

0.6 -2

-0.1 0.4 -3

0.2 -4 -0.15 0 0.05 0.1 0.15 0.2 0.25 0 x 0 1 2 3 4 5 6  o -3 z(x) & ,  = 26.18 x 10 0.15 4 C vs.  L 3 3 0.1

2 2.5

0.05 1 2

0 0 z(x)

L 1.5 C -1 -0.05

1 -2

-0.1 -3 0.5

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 0 x 0 1 2 3 4 5 6 7 8  o -3 z(x) & ,  = 29.98 x 10 0.15 4 C vs.  L 4 3 0.1

2 3 0.05 1 2

0 0 z(x)

L 1 C -1 -0.05

0 -2

-0.1 -3 -1

-4 -0.15 0 0.05 0.1 0.15 0.2 0.25 -2 x 0 1 2 3 4 5 6 7 8 9 

Figure A-7: Increasing angle of attack showed good data until stall, at which the TEV interaction caused bad lift coefficient data.

161

C vs.  L 20

10 L C

-5 0 5 10 15 20 25 

Figure A-9: Lines of code showing the improper location of the TEW_delta_z line.

162

Figure A-10: Lines of code showing the corrected location of the TEW_delta_z line.

C vs.  L 3

2.5

L 1.5 C

0.5

0 0 2 4 6 8 10 12 14  Figure A-11: CL vs. convective time using the Uhlman method after correcting the location of the TEW_delta_z line of code.

163