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Stall, Flutter and Thrust Generation of an Oscillating Airfoil

VAITLA LAXMAN vaitla.laxman@gmail.com

Department of Aerospace Engineering, Amrita School of Engineering, Amrita University, Coimbatore

Pravartana 2016, 12-02-2016 IIT Kanpur

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Outline of the presentation

• Introduction

• Objectives of the study • Dynamic stall model

• Determination of Flutter

• Generation of Thrust – (Harmonic vs non-harmonic periodic motion)

• Conclusions

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Introduction

• Oscillating airfoil has two types of motions (i) pitching, and (ii) plunging/heaving motions

• Dynamic stall: Heaving and pitching motions increase the angle of attack airfoil stalled due to high angle of attack in dynamically

• Flutter: Sufficient damping in the system when system is in

unsteady environment damping changes from positive to negative via zero (flutter boundary)

• These two phenomenon are not desirable but unavaoidable – Is there any simple model to predict these two phenomenon?

• Flapping wings Generation of thrust nonSHM-periodic motion

– SHM vs nonSHM which motion is good for generation of thrust?

Objectives of this study

• Objective is to develop a unsteady aerodynamic model (In differential equations forms)

• Focus on dynamic stall model • Focus on flutter prediction

• Systematic analysis to identify the effects of different plunging

profiles on generation of thrust

• Parametric study on the effect of heave amplitude and reduced frequency on generation of thrust

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Dynamic stall model

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• UNSTEADY AERODYNAMICS (THEODORSEN’S THEORY - 1935)

AERODYNAMIC MODELS

• Lift has two components (i) Noncirculatory lift and (ii) Circulatory lift

• C(k) is Theodorsen’s lift deficiency function, C(k) = F(k) + iG(k)

• Where k is called reduced frequency (ωb/V)

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ONERA EDLin Extended Model (1995)

• Lift at quarter-chord point:

( )

2 2 2

0

0 1 1 2

1 1 0 1

0 1

2

2

0

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L

L

wzV

L S sbW kbW V V

CzV V VW Wb b b

Cz d W W

V Va rb b

V Vr V C E Wb b

ρ

λ λ λθ

α ασθ

= + + Γ + Γ

∂ Γ + Γ = + ∂ ∂ + + + ∂

Γ + Γ + Γ = − ∆ +

0

1

,

( );

where

hW VV

W h

θ

θ

= +

=

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• For an airfoil oscillating under unstalled conditions, lift equation can be simplified as

• This expression is identical to the unsteady lift obtained by Theodorsen (1935), if CP(k) is replaced by C(k)

( )0 1 0 11 1 ( ) 2 22 2 2 PL S bW bW SVC k W Wπρ π ρ π π = + + +

( )P

b SVC kb SV

α λ

λ

+ = +

Petot’s approximate function to C(k) S=iω

Theodorsen Model

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( )

22

1 2 3

22

2 3

'

b bA S A S AV VC k

b bS B S BV V

+ + = + +

1 2 3 2 30.50; 0.393; 0.0439; 0.5515; 0.0439;A A A B B= = = = =

( )20.53 0.25* (1 ) 1 ; 0.17 0.13* ;M Mα λ= + − − = −

• Petot’s approximate function to C(k):

• Second order approximate function to C(k):

Venkatesan and Friedmann (1986)

Rational Approximation

( )P

b SVC kb SV

α λ

λ

+ = +

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• Comparison between Petot approximate function and Second order

approximate function to C(k)

Comparison of Lift Deficiency Function

Exact Theodorsen’s Function Second Order Approximation Petot Approximate Function

Exact Theodorsen’s Function Second Order Approximation Petot Approximate Function

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• Replacing first order approximation by the second order rational approximation and applying Laplace inverse transform, the Modified lift equations are obtained as:

Modified Stall Model

( )

02 2 2

0 1 1 2

2 2 2

1 2 1 3 1 3 0 3 1

2 0 2 1 1 0 1 1

2 2

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L

L L

wzV

L S sbW kbW V V

CzV V V VB B A W A Wb b b b

Cz CzV VA W A W A W A Wb b

V V Va r r V Cb b b

ρ

σθ

σ σθ θ

= + + Γ + Γ

∂ Γ + Γ + Γ = + ∂ ∂ ∂ + + + + ∂ ∂

Γ + Γ + Γ = − ∆ +

0

VE Wb

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( ) ( )0

0

:

15 ; 10 ;0.05 & 0.1;0.2 ; 0.3

o o

Pitch

t Cos t

k kb m M

θ θ θ ω

θ θ

= +

= =

= == =

• An airfoil undergoing only pitching motion

Sample result: Pitching Motion

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• Petot stall model

• Modified stall model

Pitching Motion (Cont’d)

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k=0.03 k=0.05 k=0.1

Pitching Motion (Cont’d)

Lift

Moment

Drag

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Flutter Prediction

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• Equations of Motion:

Equation for flutter

;

;hmh S K h L

I S h K Mφ

φ φ φ

φ

φ φ

+ + = −

+ + =

METHODS OF DETERMINING FLUTTER BOUNDARY

• p-method (Quasi Static aerodynamics)

• U-g method (Theodorsen's unsteady aerodynamics)

• p-k method (Theodorsen's unsteady aerodynamics)

• State space method (ONERA model)

;

;hmh S K h L

I S h K Mφ

φ φ φ

φ

φ φ

+ + = −

+ + =

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p-method

• Clear prediction but results are not accurate • Neglects unsteady effects

U-g Method

• Introduction of artificial damping does not reflect the real phenomenon • Does not predict system response except at flutter point • Predicts the incorrect mode (torsion) that becomes unstable

P-k Method

• Predicts response of the system correctly at all velocities

• Predicts the correct mode (bending) that becomes unstable

ONERA Method

• The C(k) term involved is of first order approximation which is less accurate

Modified ONERA Method

• The C(k) term involved is of second order approximation which

• Can be extended to wing flutter

METHODS OF DETERMINING FLUTTER BOUNDARY

MODELS METHOD USED

FLUTTER VELOCITY

(m/s) Quasi-Static p-method 70-90

Quasi-Steady p-method 70-80 Theodorsen’s unsteady

Aerodynamic theory U-g (k) method

92.92

Theodorsen’s unsteady Aerodynamic theory

p-k method 91

ONERA model State space 98

Modified ONERA model

State space 100

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Thrust Generation of an Oscillating Airfoil

Introduction - Literature Study

• Research has been focused on flapping-wing aerodynamics understand the generation of thrust due to oscillating airfoil

• However, research is focused on an airfoil undergoing SHM

• Contribution of plunging motion towards propulsion efficiency is much higher than that of pitching motion (Read et al)

• Numerical simulation is attempted to identify the effects of different

plunging profiles on generation of thrust

• Parametric study on the effect of heave amplitude and reduced frequency on generation of thrust

Numerical simulation

• FLUENT is used to simulate the unsteady incompressible laminar viscous flow field around the plunging airfoil

• The working fluid is considered to be water

• Reynold’s number is 20,000

• Solution convergence is maintained to be 10-5

Generation of the Grid • Combination of three mesh; Inner fine mesh around airfoil • Intermediate fine mesh around inner fine mesh • Deforming outer mesh around intermediate mesh

Generation of the Grid cont’d

• Inner fine mesh is created by CONSTRUCT 2D (open source code) • Intermediate fine mesh around inner fine mesh - GAMBIT • Deforming outer mesh around intermediate mesh - GAMBIT • The inner meshes are made to plunge as a rigid body • The plunge motion is incorporated by user defined function (UDF)

Various plunging profiles

Influence of grid and time step • Effect of grid • Effect of time steps

• Refined mesh (1.26 x 105 ) is used for further simulations

• 800 time steps for a cycle is chosen for further simulations

Validation of the model

• For plunging airfoil NACA 0012 at heaving amplitude 0.175

• Various of CT and CP results are in close agreement with experimental data

Validation of the model cont’d

• For plunging airfoil NACA 0012 for reduced frequency 1

• Various of CT with plunging amplitude is in good agreement with numerical simulation

Results and discussion • Effect of reduced frequency (k)

• For plunging amplitude 0.25 • For plunging amplitude 1

• Effect of plunging amplitude (h)

Results and discussion cont’d

• For reduced frequency 1 • For reduced frequency 5

Conclusions • Modified dynamic stall model

– predicts variation of the lift in attached and stalled region – predicts flutter boundary – Can be extended to predict wing flutter under going

• Thrust generated by the square and trapezoidal (periodic)

plunging motions is much higher than the sinusoidal plunging motion

• At higher reduced frequency and amplitudes, trapezoidal plunging motion generates higher thrust

• Analytical model for lift for any periodic motion under going

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Thank you!

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• Lift variation for pitching and plunging motion

Pitching and Plunging Motion (Cont’d)

• Pitching Motion • Plunging Motion

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• Moment variation for pitching and plunging motion

Pitching and Plunging Motion (Cont’d)

• Pitching Motion • Plunging Motion