unsteady lift of thick airfoils in incompressible turbulent flow

154 Pages·2011·2.29 MB·English

The Pennsylvania State University
The Graduate School
Graduate Program in Acoustics
UNSTEADY LIFT OF THICK AIRFOILS
IN INCOMPRESSIBLE TURBULENT FLOW
A Dissertation in
Acoustics
by
Peter D. Lysak
c 2011 Peter D. Lysak
(cid:13)
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2011
The dissertation of Peter D. Lysak was reviewed and approved by the following:
∗
Dean E. Capone
Sr. Research Associate and Associate Professor of Acoustics
Dissertation Adviser
Chair of Committee
Kenneth S. Brentner
Professor of Aerospace Engineering
Michael L. Jonson
Research Associate
Special Member
Philip J. Morris
Boeing / A. D. Welliver Professor of Aerospace Engineering
Victor W. Sparrow
Professor of Acoustics
Interim Head of the Graduate Program in Acoustics
Signatures are on file in the Graduate School.
∗
Abstract
The unsteady lift forces that act on an airfoil in turbulent flow are an undesirable source of
vibration and noise in many industrial applications. Methods to predict these forces have
traditionally treated the airfoil as a flat plate. At higher frequencies, where the relevant
turbulent length scales are comparable to the airfoil thickness, the flat plate approximation
becomes invalid and results in overprediction of the unsteady force spectrum. This work
provides an improved methodology for the prediction of the unsteady lift forces that ac-
counts for the thickness of the airfoil. An analytical model was developed to calculate the
response of the airfoil to high frequency gusts. The approach is based on a time-domain
calculation with a sharp-edged gust and accounts for the distortion of the gust by the mean
flowaroundtheairfoilleadingedge. Theunsteadyliftiscalculatedfromaweightedintegra-
tion of the gust vorticity, which makes the model relatively straightforward to implement
and verify. For routine design calculations of turbulence-induced forces, a closed-form gust
response thickness correction factor was developed for NACA 65 series airfoils. The model
was then validated by measuring the unsteady lift spectrum using piezoelectric force gages
in a water tunnel with grid-generated turbulence. A series of four airfoils with thickness-
to-chord ratios ranging from 8 to 16 percent were tested over a wide range of speeds. In
addition, the turbulence spectrum was measured using Laser Doppler Velocimetry. The ex-
perimental results confirmed that the analytical model accurately predicts the attenuation
of the high frequency gust response due to the airfoil thickness.
iii
Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction 1
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Background 9
2.1 Unsteady Lift from Inflow Turbulence . . . . . . . . . . . . . . . . . . . . . 10
2.2 Gust Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Discrete Vortex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Unsteady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Unsteady Pressure and Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Sinusoidal Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Three-Dimensional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Analytical Model 37
3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Vorticity Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Unsteady Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.3 Step Function Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Steady Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.2 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
iv
3.3.3 Joukowski Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.4 Von K´arm´an-Trefftz Airfoil . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.5 Stagnation Streamline . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Drift Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 Drift Function Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Unsteady Lift and Frequency Response . . . . . . . . . . . . . . . . . . . . . 66
4 Model Predictions 70
4.1 Flat Plate Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.1 Spatial Truncation Criteria . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.2 Viscous Diffusion and Boundary Layer Effects . . . . . . . . . . . . . 76
4.2 Gust Upwash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Effect of Airfoil Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Effect of Camber and Angle of Attack . . . . . . . . . . . . . . . . . . . . . 83
4.5 Effect of Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Experimental Validation 94
5.1 Test Facility and Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Turbulent Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.1 Speed Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.3 Correlation Functions and Integral Length Scales . . . . . . . . . . . 102
5.2.4 Velocity Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Unsteady Force Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1 System Resonances and Calibration . . . . . . . . . . . . . . . . . . 108
5.3.2 Data Processing and Background Noise Removal . . . . . . . . . . . 111
5.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Conclusion 121
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 122
A Stochastic Modeling of Turbulence 124
A.1 Turbulence Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.2 Spatial Correlation and Integral Length Scale . . . . . . . . . . . . . . . . . 125
A.3 Wavenumber Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . 126
v
A.4 Turbulence Spectrum Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.5 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.6 Frequency Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.7 Spanwise Correlation Length . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B Gust Response Phase 131
Bibliography 136
vi
List of Tables
3.1 Von K´arm´an-Trefftz airfoil parameters. . . . . . . . . . . . . . . . . . . . . . 60
4.1 Airfoil parameters and gust response attenuation coefficients. . . . . . . . . 92
5.1 LDV measured mean and rms velocity profiles. . . . . . . . . . . . . . . . . 102
5.2 Integral length scales determined from LDV correlation measurements. . . . 105
5.3 Natural frequencies of the first bending modes for airfoils in air and in water.109
B.1 Best fit constants for the phase slope found by linear regression. . . . . . . 132
vii
List of Figures
2.1 Sinusoidal gust with streamwise and spanwise variation. . . . . . . . . . . . 10
2.2 Discrete vortex model for a two-dimensional flat plate airfoil. . . . . . . . . 15
2.3 Steady flow solution for a two-dimensional flat plate airfoil. . . . . . . . . . 17
2.4 Initial flow solution for a two-dimensional flat plate airfoil. . . . . . . . . . . 19
2.5 Unsteady two-dimensional discrete vortex model with vortex wake. . . . . . 20
2.6 Transient solution to the suddenly imposed constant gust problem. . . . . . 21
2.7 Transient solution to the step function gust problem. . . . . . . . . . . . . . 22
2.8 Solution to the impulsive gust problem. . . . . . . . . . . . . . . . . . . . . 23
2.9 Unsteady lift for a suddenly imposed constant gust and step function gust. 25
2.10 Frequency response function obtained from the impulse function gust solution. 26
2.11 Frequency response function obtained from the impulse function gust solu-
tion without the vortex wake. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.12 Discrete vortex method solution for a sinusoidal gust. . . . . . . . . . . . . 30
2.13 Panel arrangement for the three-dimensional gust response. . . . . . . . . . 32
2.14 Image panels for endwall boundary conditions. . . . . . . . . . . . . . . . . 34
2.15 Doublet source strengths with and without endwalls at three time steps. . . 35
2.16 Frequency response function for a three-dimensional gust. . . . . . . . . . . 36
3.1 Step function gust. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Volume of fluid used in deriving the unsteady lift formula. . . . . . . . . . . 43
3.3 Geometry used to reduce the area integral to a line integral. . . . . . . . . . 47
3.4 Step function gust vorticity in three dimensions. . . . . . . . . . . . . . . . 49
3.5 NACA 65 -012 airfoil compared to a 12% thick ellipse. . . . . . . . . . . . 52
1A
3.6 Streamlines about a circle and 12% thick ellipse. . . . . . . . . . . . . . . . 55
3.7 Circle plane geometry used in the Joukowski transformation. . . . . . . . . 56
3.8 Comparison of Joukowski and von K´arm´an-Trefftz airfoils. . . . . . . . . . . 59
3.9 Drift lines of the flow around a 12% thick ellipse. . . . . . . . . . . . . . . . 63
3.10 Schematic used to determine the normal to a curve using finite differences.. 64
3.11 Unsteady lift generated by a step function gust. . . . . . . . . . . . . . . . . 67
viii
3.12 Effect of thickness on gust frequency response. . . . . . . . . . . . . . . . . 69
4.1 Unsteady lift produced by a step function gust acting on a flat plate airfoil. 72
4.2 Flat plate impulse response and window tapering function. . . . . . . . . . 73
4.3 Flat plate frequency response calculated from the tapered impulse response. 73
4.4 Effect of window tapering function. . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Integrand of the unsteady lift integral. . . . . . . . . . . . . . . . . . . . . . 75
4.6 Truncation error due to the finite integration domain. . . . . . . . . . . . . 76
4.7 Effect of the viscous boundary layer on unsteady lift. . . . . . . . . . . . . . 78
4.8 Upwash induced by a distorted step function gust. . . . . . . . . . . . . . . 81
4.9 Upwash impulse associated with the distorted step function gust. . . . . . . 81
4.10 Upwash frequency spectrum associated with the distorted step function gust. 82
4.11 Gust response for a von K´arm´an-Trefftz airfoil and an ellipse. . . . . . . . . 83
4.12 Gust response for von K´arm´an-Trefftz and Joukowski airfoils of the same
thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.13 Gust response for a 12% thick von K´arm´an-Trefftz airfoil and a 9% thick
Joukowski airfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.14 Comparisonof12%thickvonK´arm´an-Trefftzairfoiland9%thickJoukowski
airfoil geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.15 Streamlines about a 12% thick airfoil at 5 angle of attack. . . . . . . . . . 85
◦
4.16 Gust response for an airfoil at angle of attack. . . . . . . . . . . . . . . . . . 86
4.17 Streamlines about a 12% thick cambered airfoil at 5 angle of attack. . . . . 87
◦
4.18 Gust response for a cambered airfoil. . . . . . . . . . . . . . . . . . . . . . . 87
4.19 Drift line near the airfoil leading edge. . . . . . . . . . . . . . . . . . . . . . 88
4.20 Gust response for an 8% thick von K´arm´an-Trefftz airfoil. . . . . . . . . . . 89
4.21 Gust response for a 10% thick von K´arm´an-Trefftz airfoil. . . . . . . . . . . 90
4.22 Gust response for a 12.5% thick von K´arm´an-Trefftz airfoil. . . . . . . . . . 90
4.23 Gust response for a 16% thick von K´arm´an-Trefftz airfoil. . . . . . . . . . . 91
4.24 Gust response attenuation coefficient as a function of airfoil thickness. . . . 93
5.1 Photograph of the water tunnel test section. . . . . . . . . . . . . . . . . . . 95
5.2 Side view of the water tunnel test section. . . . . . . . . . . . . . . . . . . . 96
5.3 Photograph of the turbulence-generating grid next to two airfoils. . . . . . . 97
5.4 Photograph of the water tunnel test section with airfoil installed. . . . . . . 97
5.5 Schematic of the water tunnel test section hardware. . . . . . . . . . . . . . 98
5.6 Photograph of the LDV system. . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7 Photograph of the LDV equipment and traversing table. . . . . . . . . . . . 100
ix
5.8 Water tunnel speed calibration with turbulence grid installed. . . . . . . . . 101
5.9 Measured longitudinal correlation function compared to model. . . . . . . . 104
5.10 Measured transverse correlation function compared to model. . . . . . . . . 105
5.11 Measured longitudinal velocity spectrum compared to model. . . . . . . . . 107
5.12 Measured transverse velocity spectrum compared to model. . . . . . . . . . 107
5.13 Predicted mode shape of the first bending mode for an airfoil in water. . . . 110
5.14 Calibration of the summed force gage response. . . . . . . . . . . . . . . . . 111
5.15 Example results of the coherent noise removal technique. . . . . . . . . . . . 113
5.16 Example signal-to-noise results. . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.17 Collapse of the measured force spectra to a single non-dimensional curve. . 115
5.18 Background noise correction applied to the 12.5% thick airfoil. . . . . . . . 116
5.19 Background noise correction applied to the 16% thick airfoil. . . . . . . . . 116
5.20 Repeatability of the measured force spectra. . . . . . . . . . . . . . . . . . . 117
5.21 Force spectrum results for the 8% thick airfoil. . . . . . . . . . . . . . . . . 118
5.22 Force spectrum results for the 10% thick airfoil. . . . . . . . . . . . . . . . . 119
5.23 Force spectrum results for the 12.5% thick airfoil. . . . . . . . . . . . . . . . 119
5.24 Force spectrum results for the 16% thick airfoil. . . . . . . . . . . . . . . . . 120
A.1 Longitudinal and transverse correlations. . . . . . . . . . . . . . . . . . . . . 126
B.1 Curve fit for phase slope parameter a. . . . . . . . . . . . . . . . . . . . . . 133
B.2 Curve fit for phase slope parameter b. . . . . . . . . . . . . . . . . . . . . . 133
B.3 Refined curve fit for phase slope parameter a. . . . . . . . . . . . . . . . . . 134
B.4 Gust response phase for the 12% thick airfoil. . . . . . . . . . . . . . . . . . 135
x