Optimization of Flapping-Flight Using Numerical ...

Optimization of Flapping-Flight Using Numerical Lifting-Line Analysis

Doug Hunsaker, Ph.D.! Utah State University! Logan, UT

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Introduction ❖

Undergraduate - Brigham Young University! ❖

Aerovironment!

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MAGICC Lab!

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Masters - Brigham Young University! ❖

Thesis: Aerodynamics of VTOL Aircraft!

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PhD - Utah State University! ❖

Dissertation: Turbulence Modeling!

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Sandia National Labs: CACTUS

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Introduction ❖

Scaled Composites! ❖

SpaceShipTwo, WhiteKnightTwo: Flutter!

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Stratolaunch: Aerodynamics /Aeroelastic Loads!

virgingalactic.com

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stratolaunch.com

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Post-Graduation Publications!

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Phillips, W. F., and Hunsaker, D. F., “Lifting-Line Predictions for Induced Drag and Lift in Ground Effect,” Journal of Aircraft, Vol. 50, No. 4, pp. 1226 – 1233, July – August 2013.!

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Hunsaker, D. F., and Phillips, W. F., “Momentum Theory with Slipstream Rotation Applied to Wind Turbines,” 31st AIAA Applied Aerodynamics Conference, San Diego, California, June 24 – 27, 2013, AIAA-2013-3161.!

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Phillips, W. F., and Hunsaker, D. F., “Lifting-Line Predictions for Induced Drag and Lift in Ground Effect,” 31st AIAA Applied Aerodynamics Conference, San Diego, California, June 24 – 27, 2013, AIAA-2013-2917.! Phillips, W. F., Miller, R. A., and Hunsaker, D. F., “Decomposed Lifting-Line Predictions and Optimization for Propulsive Efficiency of Flapping Wings,” 31st AIAA Applied Aerodynamics Conference, San Diego, California, June 24 – 27, 2013, AIAA-2013-2921.! Phillips, W. F., Fowler, E.B., and Hunsaker, D. F., “Energy-Vorticity Turbulence Model with Application to Flow near Rough Surfaces,” AIAA Journal, Vol. 51, No. 5, pp. 1211 – 1220, May 2013.

Outline ❖

Why Flapping Flight!

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Flight Scales!

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Numerical Lifting-Line Theory!

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Optimization!

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Quasi-Steady Results!

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Future Work / Collaboration

Why Flapping Flight? ❖

It’s Natural!

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It’s Efficient!

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It’s Versatile!

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It’s Quiet!

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It’s Illusive

Flight Scales 1.0E+00'

Potential Flow with! Boundary Layer! Theory!

Unsteady

Time Dependency

1.0E%01'

2D limit (Theodorsen) Quasi-Steady

Reduced Frequency Reduced&Frequency&

!

Reduced! Frequency

N-S Needed

1.0E%02' 1.0E+02'

Reasonably Accurate

1.0E+03'

1.0E+04'

Chord*Frequency! Velocity

Very Accurate 1.0E+05'

Reynolds&Number& Reynolds Number

1.0E+06'

1.0E+07'

segment of one horseshoe vortex and the right-hand trailing segment of the next. This is N ΓIn v for display purposes only. the left-hand corner of one horseshoe and the rightj reality, ji ~ ~ = + V V ∞ next are both placed at the same point. Thus, except at(1.9.4) i hand corner of the the wingtips, C L ( z ) = C L,α [α eff ( z ) − α L 0 ( z )] c each trailing vortex segment j =1 isjcoincident with another trailing segment from the adjacent vortex. If two adjacent vortices have exactly the same strength, then the two coincident where αeff is the local section angle of attack, including the effects of velocity induced by trailing segments exactly cancel, since one has clockwise rotation and the other has the trailing vortex sheet. Because of the downwash induced on the wing by trailing V∞the is the velocity of the uniform flow, is net thevorticity strength horseshoe vortex j, where counterclockwise rotation.ΓjThe that isof shed from the wing at any internal vortex sheet, the local relative wind is inclined at an angle, αi, to the freestream, as shown is simply the difference in strength the two adjacent share the node. v ji is velocitynode that would be induced atofcontrol pointvortices i bythat horseshoe in Fig. 1.8.6. This angle is called the induced angle of attack. Since the aliftdimensionless is always Each horseshoe vortex is composed of three straight segments, a finite bound j, having perpendicular to the local relative wind, the downwash tilts the liftvortex vector back, creating aa unit strength segment and two semi-infinite trailing segments. From Eqs. (1.5.18) and (1.5.19), we can component of lift parallel to the freestream. This is called induced drag. calculate the velocity induced at an arbitrary point in space (x,y, z) by a general horseshoe When the downwash is accounted for, Prandtl’s hypothesis requires that vortex. As shown in Fig. 1.9.3, a general horseshoe vortex is completely defined by two Chapter 1 Overview of Aerodynamics ⎧ cj ⎡ u ∞ × r j2i nodal points, r j(x1i 2×,y2r,zj22),i )a trailing unituvector, (r j1(xi 1+,y1r,zj21i) )( and ∞ × r ju1i∞ , and ⎤a vortex strength, Γ. ~ j≠i + − b/2 The horseshoe vortex starts at the fluid boundary, an infinite distance The ⎪ ⎢ r (r − u ⋅ r ) r r (r r + r ⋅ r ) r (r − ⎥ ,downstream. 2Γ ( z ) C L,α ~ 1 d Γ ⎛ ⎞ π u r 4 ) ⋅ j i j i ∞ j i j i j i j i j i j i j i j i j i ∞ j i ⎪ + d ζ = C [ α ( z ) − α ( z )] (1.8.2) 2 2 2 1 2 1 2 1 2 1 1 1 ⎜ ⎟ L ,α inbound trailing vortex segment is directed along the vector −u to⎦node 1 at (x ,y1,z1). ⎣ ∫ approximate The model usedV∞by c( z )Prandtl 4πV∞ ζ =to z − ζ ⎝ dz ⎠ z =ζ the bound vorticity vLji0 = and ⎨ trailing vortex The bound vortex segment is directed along the wing lifting line∞ from node 1 to1node −b / 2 u ∞wing u ∞outbound × r j2i at (x ,y ,z ). The × r j1i trailing ⎤ vortex segment is directed along the vector u from2 is shown in Fig. 1.8.5. All bound vortex filaments are assumed to the j ⎡ ⎪ cfollow 2− 2 2 , j =i ∞ ⎥infinite node 2 to the fluid boundary, an distance back downstream. The velocity induced er-chord line, and all trailing vortex filaments are assumed to be straight parallel ⎪⎩ 4π ⎢⎣ rand u r u r ( r ) r ( r ) − ⋅ − ⋅ j2i j2i ∞ j2i j1i j1i ∞ j1i ⎦ by the entire horseshoe vortex is simply the vector sum of the velocities induced by each he freestream. Rollup of the trailing vortex sheet is ignored. of the Chapter three linear segments that make up horseshoe. (1.9.5) 1 Overview oftheAerodynamics

∑

Lifting-Line Theory

Prandtl - 1921

Phillips and Snyder - 2000

100

The foundation of lifting-line theory is the requirement that for each cross-section of ~ ing, the lift predicted from the vortex lifting law must to that predicted the spatial vector from horseshoe vortex j to the control point of r j1ibeisequal C Li , 1αof wherenode i, and δi are, respectively, the local airfoil section lift airfoil section theory, i.e., horseshoe vortex i, r j2i is angle the spatial vector node flap 2 ofdeflection, horseshoe all vortex j to the of attack, andfrom the local evaluated for the a control point of horseshoewith vortex i, and u ∞ i.isDefining the unit uvector in the direction of the control point ~ ~ ni and u ai to be the local unit norm C L ( z ) = C L,α [α eff ( z ) − α L 0 ( z )]freestream. At this point, c j could be any characteristic length associated with the wing for the airfoil section located at control point i, as shown in Fig. 1.9 section aligned with horseshoe j. point This characteristic attackvortex at control i can be writtenlength as is simply used to e αeff is the local section angle of attack, including the effects of velocity induced by and has no effect on the induced velocity. An appropriate nondimensionalize Eq. (1.9.5) ailing vortex sheet. Because of the downwash inducedchoice on thefor wing c j by willthe betrailing addressed at a later point. The bound vortex segment is excluded ⎛ Vi ⋅ u ni ⎞ x sheet, the local relative wind is inclined at an angle, αfrom as shown α i = induces tan −1 ⎜ no i, to the Eq.freestream, (1.9.5), when j = i, because a straight vortex segment downwash ⎟ V u ⋅ i ai ⎝ ⎠ g. 1.8.6. This angle is called the induced angle of attack. thelength. lift is always alongSince its own However, the second term in Eq. (1.9.5), for j ≠ i, is indeterminate ndicular to the local relative wind, the downwash tilts the lift used vector back, a ri i ri i + ri i ⋅ ri i = 0. when with j = creating i, because 1 2 1 2 If the relation implied by Eq. (1.9.7) is known at each section of the onent of lift parallel to the freestream. This is called induced drag. From Eqs. (1.8.1) and (1.9.4), the aerodynamic force acting on a spanwise differen1.9.2. Horseshoe vorticesacting distributedon overathespanwise span of a wingdifferential with sweep and dihedral. of theFigure aerodynamic force section o When the downwash is accounted for, Prandtl’s hypothesis that tial requires section of the wing located at control point i is given by control point i can be written as Figure 1.8.5. Prandtl’s model for the bound vorticity and the trailing vortex sheet generated by a ~ wing N Γ C L,α b / 2 ⎛ ⎞ 2Γof(finite z ) span. 1 ⎛ dΓ ⎞ dζ = C~ [α ( z ) − α ( z )] j 2 ~ 1 ⎜ ⎟ + (1.8.2) ⎜ ⎟ dF = ρ V C Li (α i , δ i )(1.9.6) dSi L ,α L0 ρ Γ dF = V + v × d l i ∞ ∫ i i i ji 2 V∞ c( z ) 4πV∞ ζ = − b / 2 z − ζ ⎝ dz ⎠ z =ζ ⎜ ∞ ∑ ⎟ c j =1 j ⎝ ⎠ where dSi is a spanwise differential planform area element locate Allowing for the possibility of flap the localforce section lift coefficient for theequal t Infinite Series Solution! Numerical Solution! Setting thedeflection, magnitude of the obtained from Eq. (1.9.6) airfoil section located at control point i applying is a function of localtoangle of attackand andrearranging, local flap we c Eq. (1.9.9), Eq. (1.9.4) Eq. (1.9.8), Single Wing! Multiple Wings! deflection,

Straight Quarter-Chord

Sweep, Dihedral, etc. N ~ ~ C Li = C Li (α i , δ2i⎛⎜)u + v G ⎞⎟ × ζ G − C~ (α(1.9.7) ji j i i Li i ,δ i ) = 0 ⎜ ∞ ∑ ⎟ j =1 ⎝ ⎠

Lifting-Line Theory 20"

Lifting-Line Theory!

!

Potential Flow with! Boundary Layer! Theory

Aspect Ratio Aspect'Ra*o'

15"

10"

Very Accurate

5"

Panel Codes or! Inviscid CFD Needed N-S Needed

0" 1.0E+02"

Reasonably Accurate

1.0E+03"

1.0E+04"

Very Accurate 1.0E+05"

Reynolds'Number' Reynolds Number

1.0E+06"

1.0E+07"

Lifting-Line Theory 1.0E+00'

Time Dependency!

Unsteady

Potential Flow with! Boundary Layer! Theory

1.0E%01'

2D limit (Theodorsen) Quasi-Steady

Reduced Frequency Reduced&Frequency&

!

Lifting-Line Theory! Accurate! Reasonably Accurate! Not Accurate

N-S Needed

1.0E%02' 1.0E+02'

Reasonably Accurate

1.0E+03'

1.0E+04'

Very Accurate 1.0E+05'

Reynolds&Number& Reynolds Number

1.0E+06'

1.0E+07'

Lifting-Line Theory 106

1.0

Accuracy! • •

Aspect Ratios > 4.0! Reynolds Numbers > ~100,000!

! •

Variable Chord!

•

Variable Twist!

•

Variable Plunging!

•

Unsteady Vortex Wake! was

•

0.2

0.4

0.6

0.8

1.0

1.2

1.4

McMullen/Landon

Figure 1.9.6. Comparison between the lift coefficient predicted by the numerical lifting-line method, a numerical panel method, and an inviscid CFD solutions with data obtained from wind tunnel tests for an unswept wing and a wing with 45 degrees of sweep.

The insight of Ludwig Prandtl (1875–1953) was nothing short of astonishing. This never more dramatically demonstrated than in the development of his classical lifting-line theory, during the period 1911 through 1918. The utility of this simple and Viscosity! elegant theory is so great that it is still widely used today. Furthermore, with a few minor alterations and the use of a modern computer, the model proposed by Prandtl can be used to predict the inviscid forces and moments acting on lifting surfaces of aspect ratio greater than about 4 with an accuracy as good as that obtained from modern panel codes or CFD, but at a small fraction of the computational cost. Like panel methods, lifting-line theory provides only a potential flow solution. Thus, the forces and moments computed from this method do not include viscous effects. In Optimization - Geometry Definition! addition to this restriction, which also applies to panel methods, lifting-line theory imposes an additional restriction, which does not apply to panel methods. For lifting Can be linked to structural models surfaces with low aspect ratio, Prandtl’s hypothesis breaks down and the usual relationship between local section lift and local section angle of attack no longer applies. It has long been established that lifting-line theory gives good agreement with experimental data for lifting surfaces of aspect ratio greater than about 4. For lifting

• Computationally Efficient! •

0.4

2-D Section Lift Coefficient

Variable Sweep!

!

0.6

0.0 0.0

•

•

0.8

Lifting-Line Code Panel Code CFD Code Experimental Data

0.2

Can include effects of! •

Wing Lift Coefficient

•

Chapter 1 Overview of Aerodynamics

Optimization Input Parameters! ! • Chord! • Sweep! • Plunging! • Flapping Cycle! • Twist!

Results! ! Analysis Model! ! Lifting-Line!

• Lift! • Drag! • Root Bending Moment! • Maximum Thrust! • Ideal Propulsive Efficiency
 =PAvailable/PRequired
 


Thrust*Velocity! Bending Moment*Flapping Rate

=!

Optimization Method! ! Gradient-Based! Broyden-Fletcher-Goldfarb-Shanno (BFGS)

Quasi-Steady Results ❖ ❖ ❖

Aspect Ratio = 14! Mean Lift Coefficient = 0.627! Parasitic Drag Coefficient = 0.01

Sinusoidal Flapping of an Untwisted Rectangular Wing of Aspect Ratio 14 in Pure Plunging, time/period = 0.2500

No Twist

Linear Twist

Linear Twist with the Minimum-Power Magnitude Maintained over the Flapping Cycle, time/period = 0.2500

Minimum-Power Washout Distribution-Magnitude Product Maintained over the Flapping Cycle, time/period = 0.2500

Optimum Twist

Quasi-Steady Results ❖ ❖ ❖

Aspect Ratio = 14! Mean Lift Coefficient = 0.627! Parasitic Drag Coefficient = 0.01

Sinusoidal Flapping of an Untwisted Rectangular Wing of Aspect Ratio 14 in Pure Plunging, time/period = 0.2500

No Twist: 76.5%

Linear Twist: 91.2%

Linear Twist with the Minimum-Power Magnitude Maintained over the Flapping Cycle, time/period = 0.2500

Minimum-Power Washout Distribution-Magnitude Product Maintained over the Flapping Cycle, time/period = 0.2500

Optimum Twist: 92.0%

0

5

Aerodynamic Angle of Attack (deg

5

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Aerodynamic Angle of Attack (deg

10

0.65, 0.85 0.60, 0.90 0.55, 0.95

10

0.65, 0.85 0.60, 0.90

10

z/b = 0.00, wing 10ro

0.10 0.15 5

0.20

5

Quasi-Steady Results 5

0

0.00, 0.50, 1.00 0.05, 0.45 0.10, 0.40 0.15, 0.35 0.20, 0.30 t /τ = 0.25

0

-5

0

-5

0.55, 0.95 0.00, 0.50, 1.00 0.05, 0.45 0.10, 0.40 0.15, 0.35 0.20, 0.30 t /τ = 0.25

0.25 0.30 0.35

0.40 0.45 0.50

z/b = 0.05 z/b = 0.15 z/b = 0.25 0.20.6 z/b = 0.30.8 0.41.0 0.35 z /b t /τ z/b = 0.45 1.00

0

-5

wingtip

-5 -10 -10 0.85 -50.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.0 0.1 0.5 0. 0.70, 0.80 z /b -10 0.00 0.25 0.50 0.75 t /τ =motion 0.75for a rectangular wing of aspect Figure 6. Pitching ratio 1411. withPitching sinusoidal flapping, washout,wing of a Figure motion for alinear rectangular

and the minimum-power washout magnitude.

Instantaneous Twist, ω Ω (degrees)

/b 15

Linear Twist0.5 0.4

t /τ = 0.25 0.20, 0.30 0.15, 0.35 0.10, 0.40 0.05, 0.45 0.00, 0.50, 1.00

0.0

0.2

Optimum Twist 0.4 0.6

0

t /τ 15

20

Instantaneous Twist, ω Ω (degrees)

0.3

minimum-power twist distribution–magnitude product.

15

20

z/b = 0.50, wingtip z/b = 0.40 z/b = 0.30 z/b = 0.20 t /τ = 0.25 z/b = 0.10 0.20, 0.30 z/b = 0.00, wing 0.15,root 0.35

15

twist for a rectangular wing of aspect ratio 14 with sinusoidal flap wer washout magnitude. 10

5

5

0

-5

10

0

15

0.55, 0.95 0.60, 0.90 0.65, 0.85 0.70, 0.80 t /τ = 0.75

-5

10

5

0.10, 0.40 0.05, 0.45 0.00, 0.50, 1.00

0 -5

10

5 0

z/b = 0.05 z/b = 0.15 0.55, 0.95 z/b = 0.90 0.25 0.60, 0.65, 0.85 z/b = 0.35 z/b = 0.45 0.70, 0.80t /τ = 0.75

-5

z/b = 0.00, wing root -10 -10 z/b = 0.10 z/b = 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.0 0.2 0.1 0.4 0.2 0.6 0.3 0.8 0.4 1.0 0.5 0 z /b 0.20, 0.30 t /τ z/b z = /b 0.30 t /τ = 0.25 Figure 7. Instantaneous wing twist for a rectangular wingFigure of aspect 14 with sinusoidal linear 12. ratio Instantaneous wing for a rectangular wing z/btwist =flapping, 0.40 -10

-10

Quasi-Steady Results Ideal Propulsive Efficiency

1.0

0.9

0.8

ωΩ product optimized for minimum power ω linear, Ω optimized for minimum power

0.7

pure plunging, no twist 10

15

Aspect Ratio

20

Eastern Imperial Eagle

(C) Markus Jais

Wingspan = 2 meters! Airspeed = 13.4 m/s (30 mph)! Flapping Cycle = 0.94 second! Reduced Frequency = 0.16

Wood Pigeon

(C) D. Thorns (C) Markus Jais

Wingspan = 0.75 meters! Airspeed = 15 m/s (33.5 mph)! Flapping Cycle = 0.25 second! Reduced Frequency = 0.18

Possible Future Work ❖

Quasi-Steady CFD Comparison - in progress!

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Optimize Quasi-Steady Flapping Cycle Parameters! ❖

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High-Frequency Model Development! ❖

Time-Dependent Vortex Wake!

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Optimize Unsteady Flapping Cycle Parameters!

Incorporate Aeroelastic/Inertial Effects! ❖

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Chord, Sweep, Plunging, Flapping Cycle!

Optimize Elastic/Inertial Coupling!

Proof of Concept Construction! ❖

Mechanism Design!

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Control System Design

Potential Collaborators ❖

Dr. Warren Phillips, Utah State University Emeritus! ❖

Aerodynamics and Flight Mechanics!

! ❖

Dr. Aaron Katz, Utah State University! ❖

HELIOS Developer!

! ❖

Dr. Nick Alley, Area-I! ❖

UAV SBIR/STTR!

! ❖

Scaled Composites! ❖

Rapid Prototyping

Thank You

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Backup Slides

uires pˆ rms = 0.1323, and Eqs. (41)–(43) yield

pˆ = 0.1871 sin( 2π t τ ) C L = 0.6269 + 0.4656 sin( 2π t τ )

No Twist

ults obtained from Eqs. (28)–(30) for this untwisted rectangular wing are shown in Fig. 3. The ideal propulsive ciency obtained from Eq. (37) for this flapping cycle is 76.5%. From the definition of pˆ combined with Eqs. (39) and (42), the sinusoidal flapping rate can be written as

p =

8 pˆ rms V∞ 4 pˆ rms sin( 2π t τ ) = b b

W sin( 2π t τ ) ρ SC L

(44)

s same sinusoidal flapping motion can be defined by expressing the instantaneous flapping dihedral angle, φ, in ms of its sinusoidal amplitude, φA, i.e.,

❖

φ = Efficiency φ cos( 2π t τ ) Ideal Propulsive = 76.5%

(45)

A

Sinusoidal Flapping of an Untwisted Rectangular Wing of Aspect Ratio 14 in Pure Plunging, time/period = 0.2500

ause p is the angular rate at which the dihedral angle is decreasing, from Eq. (45) we obtain

1.2

1.0 0.00 mean CDi = −0.01000

0.8 -0.02 mean CL = 0.6269

0.6 -0.04

0.4

-0.06

0.2

-0.08

CPf , Eq. (30)

Induced-Drag Coefficient

Lift and Power Coefficient

0.02

CDi , Eq. (29)

CL , Eq. (28)

mean CPf = 0.02614

0.0 0.0

-0.10

0.2

0.4

0.6

0.8

1.0

t /τ

ure 3. Variations in lift, induced-drag, and flapping-power coefficients as predicted by Eqs. (28)–(30) for usoidal flapping of an untwisted rectangular wing of aspect ratio 14. 9 American Institute of Aeronautics and Astronautics

C0 = 52.209,

C1 = 0.0,

Aerodynamic Angle of Attack (degrees)

y finding the value that gives C Di = − C D p . The mean flapping-power coefficient is then computed from integration of the results obtained from Eq. (23). 15 the time-independent linear washout and plunging distribution functions ω and ψ as specified by Eq. (56) with the same wing planform and sinusoidal lift coefficient that were used to obtain the results shown in uces

15

z/b =

t /τ = 0.75 0.70, 0.80 0.65, 0.85 0.60, 0.90 10 Linear Twist with the Minimum-Power Magnitude Maintained 0.55, 0.95

Linear Twist

C 2 = 0.0,

C L ,α Ω opt C L ,α pˆ = 0.3487 + CL CL

10

over the Flapping Cycle, time/period = 0.2500

5

5

CL , Eq. (28)

Lift and Power Coefficient

CDi , Eq. (29)

1.0 mean CDi = −0.01000

0.8 mean CL = 0.6269

0.6

0.4

0.2

CPf , Eq. (30) mean CPf = 0.02194

0.0 0.0

0.2

0.4

0.6

t /τ

0.8

0.02

0.00

-0.02

-0.04

-0.06

-0.08

Induced-Drag Coefficient

1.2

uglas Hunsaker on March 24, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2013-2921

Ideal Propulsive Efficiency = 91.2%

Instantaneous Twist, ω Ω (degrees)

ults shown in Fig. 4. Notice that by maintaining the minimum-power washout magnitude at each instant 0.00, 0.50, 1.00 0.05, 0.45 flapping cycle, we have reduced the required mean flapping-power coefficient from 0.02614 to 0.02194. 0.10, 0.40 tion in required power results from a substantial reduction in torque with only a slight reduction in 0 0 0.15, 0.35 rust for a given value of pˆ rms . To maintain airspeed and altitude with sinusoidal flapping, the rms 0.20, 0.30 t /τ = 0.25 te is increased slightly from pˆ rms = 0.1323 to pˆ rms = 0.1492, which reduces the flapping period at standard om 0.90❖s to 0.80 s. The net result is an increase in the ideal propulsive efficiency for this flapping cycle % to 91.2%. -5 -5 eriodic motion associated with the flapping cycle shown in Fig. 3 is pure plunging, i.e., the airfoil sections0.0 0.1 0.2 0.3 0.4 0.5 0.0 z /b g are not rotating in pitch during the flapping cycle. On the other hand, the flapping cycle shown in Fig. 4 ed of a combination of periodic plunging and pitching motion. As the airfoil sections of this wing plunge periodically at a rate that depends on both the spanwise coordinate and time, these sections Figure are also 6. Pitching motion for a rectangular wing of aspect rat and the minimum-power washout magnitude. pitch at a rate that varies with z and t. The instantaneous angle that the zero-lift line for each airfoil

15

15

t /τ = 0.25 0.20, 0.30 0.15, 0.35 0.10, 0.40 0.05, 0.45 0.00, 0.50, 1.00

10

5

5

0

0

0.55, 0.95 0.60, 0.90 0.65, 0.85 0.70, 0.80 t /τ = 0.75

-5

-10 -0.10

1.0

10

0.0

Lift, induced-drag, and flapping-power coefficients for a rectangular wing of aspect ratio 14 with Figure flapping, linear washout, and the minimum-power washout magnitude.

0.1

0.2

0.3

0.4

-5

-10

0.5

0.0

z /b

7. Instantaneous wing twist for a rectangular wing of a

Downloaded by Douglas Hunsaker on March 24, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.20

Aerodynamic Angle of Attack (degrees)

nts were spaced uniformly in θ. The control-point values used for the twist distribution ω (z,t)Ω (t) were d to be spanwise symmetric and 0.0 at the midspan. The (M−1)/ 2 unknown values of ω(z,t)Ω (t) at the control points were treated as independent scalar variables, to be evaluated from numerical optimization. t /τ = 0.75 used for the ω (z,t)Ω (t) product between control points were evaluated using linear interpolation in θ. At 0.70, 0.80 nt in time during the flapping cycle, the (M −1)/ 2 unknown control-point values of the ω (z,t)Ω (t) product 0.65, 0.85 10 10 11,12 0.60, 0.90 d using a computer optimization algorithm. This optimization software minimizes a single fitness Minimum-Power Washout Distribution-Magnitude Product which in this case was chosen to be the ratio defined in Eq. (51). The optimization software used to find Maintained over the Flapping Cycle, time/period = 0.2500 l-point values of the ω(z,t)Ω (t) product implements the BFGS algorithm, named after the work of 5 5 0.55, 0.95 Fletcher,14 Goldfarb,15 and Shanno.16 0.00, 0.50, 1.00 ure that the numerical solutions were grid resolved, independent solutions were obtained using coarse and 0.05, 0.45 The coarse-grid solutions used M =19 and N =39. For the fine-grid solutions M =39 and N =199 were 0.10, 0.40 0 0.15, 0.35 r the range of parameters studied, the coarse-grid solutions for propulsive efficiency were found to agree0 0.20, 0.30 ne-grid solutions to within 0.03%. For the temporal variations, both 50 and 100 time steps per cycle were t /τ = 0.25 second-order trapezoidal numerical integration. Over the range of parameters studied, the 50 time-step -5 -5 or propulsive efficiency were found to agree with the 100 time-step solutions to within 5 ×10 −10 %. this numerical optimization with sinusoidal flapping and the plunging distribution function ψ (z) for no bending given ❖ by Eq. (31), combined with the same wing planform and sinusoidal lift coefficient that were ain the results shown in Figs. 3 and 4, produces the results shown in Figs. 9–13. Comparing the flapping -10 -10 0.1 0.2 0.3 0.4 0.5 0.0 wn in Fig. 4 with that shown in Fig. 9, we find that by maintaining the minimum-power twist distribution–0.0 z /b product ω (z,t)Ω (t) at each instant during the flapping cycle, we have gained a small additional reduction ired mean flapping-power coefficient from 0.02194 to 0.02174. To maintain airspeed and altitude with flapping, the rms flapping rate is decreased slightly from pˆ rms = 0.1492 to pˆ rms = 0.1467. The netFigure result is 11. Pitching motion for a rectangular wing of aspec minimum-power twist distribution–magnitude product. rease in the ideal propulsive efficiency for this flapping cycle from 91.2% to 92.0%.

Ideal Propulsive Efficiency = 92.0%

1.2 CL

1.0 0.00 mean CDi = −0.01000

0.8 -0.02 mean CL = 0.6269

0.6 -0.04

0.4

-0.06

0.2

-0.08

C Pf mean CPf = 0.02174

0.0 0.0

-0.10

0.2

0.4

0.6

0.8

1.0

Induced-Drag Coefficient

Lift and Power Coefficient

CDi

0.02

Instantaneous Twist, ω Ω (degrees)

Optimum Twist

20

20

15

15 t /τ = 0.25 0.20, 0.30 0.15, 0.35 0.10, 0.40 0.05, 0.45 0.00, 0.50, 1.00

10 5

10 5

0

0 0.55, 0.95 0.60, 0.90 0.65, 0.85 0.70, 0.80 t /τ = 0.75

-5 -10 0.0

t /τ

Lift, induced-drag, and flapping-power coefficients for a rectangular wing of aspect ratio 14 with Figure flapping and the minimum-power twist distribution–magnitude product.

0.1

0.2

0.3

0.4

-5 -10

0.5

0.0

z /b

12. Instantaneous wing twist for a rectangular wing of a

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PhD - Turbulence Modeling

Sandia National Labs - CACTUS

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Post-Graduation Publications: Phillips, W. F., and Hunsaker, D. F., “Lifting-Line Predictions for Induced Drag and Lift in Ground Effect,” Journal of Aircraft, Vol. 50, No. 4, pp. 1226 – 1233, July – August 2013.! Hunsaker, D. F., and Phillips, W. F., “Momentum Theory with Slipstream Rotation Applied to Wind Turbines,” 31st AIAA Applied Aerodynamics Conference, San Diego, California, June 24 – 27, 2013, AIAA-2013-3161.! Phillips, W. F., and Hunsaker, D. F., “Lifting-Line Predictions for Induced Drag and Lift in Ground Effect,” 31st AIAA Applied Aerodynamics Conference, San Diego, California, June 24 – 27, 2013, AIAA-2013-2917.! Phillips, W. F., Miller, R. A., and Hunsaker, D. F., “Decomposed Lifting-Line Predictions and Optimization for Propulsive Efficiency of Flapping Wings,” 31st AIAA Applied Aerodynamics Conference, San Diego, California, June 24 – 27, 2013, AIAA-2013-2921.! Phillips, W. F., Fowler, E.B., and Hunsaker, D. F., “Energy-Vorticity Turbulence Model with Application to Flow near Rough Surfaces,” AIAA Journal, Vol. 51, No. 5, pp. 1211 – 1220, May 2013.

Flight Scales 1.0E+00'

Time Dependency!

Unsteady

Potential Flow with! Boundary Layer! Theory!

!

Lifting-Line Theory! Accurate

1.0E%01'

2D limit (Theodorsen) Quasi-Steady

Reduced Frequency Reduced&Frequency&

!

Reduced! Frequency

N-S Needed

1.0E%02' 1.0E+02'

Reasonably Accurate

1.0E+03'

1.0E+04'

Chord*Frequency! Velocity

Very Accurate 1.0E+05'

Reynolds&Number& Reynolds Number

1.0E+06'

1.0E+07'