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 1 IIT Kanpur
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 2
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? 3
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
Dynamic stall model 5
AERODYNAMIC MODELS • UNSTEADY AERODYNAMICS (THEODORSEN’S THEORY - 1935) • 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 ( ω b/V ) 6
ONERA EDLin Extended Model (1995) • Lift at quarter-chord point: ( ) 1 = ρ + + Γ + Γ L S sbW kbW V V 0 1 1 2 2 ∂ V V Cz V Γ + λ Γ = λ + λ L W W ∂ θ 1 1 0 1 b b b ∂ Cz + α + + ασ L d W W ∂ θ 0 1 2 V V where , Γ + Γ + Γ = a r 2 b 2 b 2 h = θ + ( ); W V 2 0 V V V − ∆ + r V C E W w 0 = θ z 0 b b W h V 1 7
Theodorsen Model • For an airfoil oscillating under unstalled conditions, lift equation can be simplified as π 1 1 ( ) = ρ π + + ρ π + π L S bW bW SVC ( ) 2 k W 2 W 0 1 P 0 1 2 2 2 • This expression is identical to the unsteady lift obtained by Theodorsen (1935), if C P (k) is replaced by C(k) b α + λ S ( ) V = Petot’s approximate function to C(k) C k P b S=i ω + λ S V 8
Rational Approximation • Petot’s approximate function to C(k): b α + λ S ( ) V = C k P b + λ S V ) ( α = + − − λ = − 2 0.53 0.25* (1 M ) 1 ; 0.17 0.13* M ; • Second order approximate function to C(k): 2 b b + + 2 A S A S A 1 2 3 ( ) V V = Venkatesan and Friedmann (1986) C ' k 2 b b + + 2 S B S B 2 3 V V = = = = = 0.50; 0.393; 0.0439; 0.5515; 0.0439; A A A B B 1 2 3 2 3 9
Comparison of Lift Deficiency Function • Comparison between Petot approximate function and Second order approximate function to C(k) Exact Theodorsen’s Function Second Order Approximation Petot Approximate Function Exact Theodorsen’s Function Second Order Approximation Petot Approximate Function 10
Modified Stall Model • Replacing first order approximation by the second order rational approximation and applying Laplace inverse transform, the Modified lift equations are obtained as: ( ) 1 = ρ + + Γ + Γ L S sbW kbW V V 0 1 1 2 2 2 2 ∂ 2 V V V Cz V Γ + Γ + Γ = + σ L B B A W A W ∂ θ 1 2 1 3 1 3 0 3 1 b b b b ∂ ∂ V Cz V Cz + + σ + + σ L L A W A W A W A W ∂ θ ∂ θ 2 0 2 1 1 0 1 1 b b 2 2 V V V V Γ + Γ + Γ = − ∆ + a r r V C E W w z 0 0 2 b 2 b 2 b b V 11
Sample result: Pitching Motion • An airfoil undergoing only pitching motion Pitch : ( ) ( ) θ = θ + θ ω t Cos t 0 θ = θ = o o 15 ; 10 ; 0 = = k 0.05 & k 0.1; = = 0.2 ; 0.3 b m M 12
Pitching Motion (Cont’d) • Modified stall model • Petot stall model 13
Pitching Motion (Cont’d) k=0.03 k=0.05 k=0.1 Lift Moment Drag 14
Flutter Prediction 15
Equation for flutter • Equations of Motion: + φ + = − mh S K h L ; φ h φ + + φ = I S h K M ; φ φ φ 16
METHODS OF DETERMINING FLUTTER BOUNDARY + φ + = − mh S K h L ; φ h φ + + φ = I S h K M ; φ φ φ p -method (Quasi Static aerodynamics) • • U- g method (Theodorsen's unsteady aerodynamics) p - k method (Theodorsen's unsteady aerodynamics) • • State space method (ONERA model)
p-method • Clear prediction but results are not accurate • Neglects unsteady effects 18
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 FLUTTER USED VELOCITY (m/s) Quasi-Static p -method 70-90 Quasi-Steady p -method 70-80 Theodorsen’s unsteady U- g ( k) 92.92 Aerodynamic theory method Theodorsen’s unsteady p-k method 91 Aerodynamic theory ONERA model State space 98 Modified ONERA model State space 100
Thrust Generation of an Oscillating Airfoil 24
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 10 5 ) 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 C T and C P 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 C T 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
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