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The Vortex Lattice Method . for the Rotor-Vortex Interaction Problem

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The Vortex Lattice Method . for the Rotor-Vortex Interaction Problem NASA CONTRACTOR NASA CR-2421 REPORT - THE VORTEX LATTICE METHOD . FOR THE ROTOR-VORTEX INTERACTION PROBLEM by Raghuveera Padakannaya Prepared by THE PENNSYL VANIA ST ATE UNIVERSITY University Park, Pa. 16802 for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY 1974 1. Report No. , I 2. Government Accession No. 3. Recipient's Catalog No. CR-Z4Z1 4. TItle and Subtitle 5. Report Date THE VORTEX LATTICE METHOD FOR THE ROTOR-VORTEX JULY 1974 INTERACTION PROBLEM 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. Raghuveera Padakannaya 1-------------------------------1 10. Work Unit No. g. Performing Organization Name and Address The Pennsylvania State University 11. Contract or Grant No. University Park, PA 16802 NGR - 39 - 009 - 111 ~------------------------------I13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Contractor Report National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, D.C. 20546 15. Supplementary Notes Topical report 16. Abstract This study is concerned with the rotor blade-vortex interaction problem and the resulting impul sive airloads which generate undesirable noi se level s. A numerical 1ifting surface method to predict unsteady aerodynamic forces induced on a finite aspect ratio rectangular wing by a straight, free vortex placed at an arbitrary angle in a subsonic incompressible free stream is developed first. Using a rigid wake assumption, the wake vortices are assumed to move down­ steam with the free steam velocity. Unsteady load.distributions are obtained which compare favorably with the results of planar lifting surface theory. The vortex lattice method has been extended to a single bladed rotor operating at high advance 'ratios aodeocc)untering a ft:eevottex from afixed.wing upstream of the rotor. The pre­ dicted unsteady load distributions on the model rotor blade are generally in agreement with the experimental results. This method has also been extended to full scale rotor flight cases in which vortex induced loads near the tip of a rotor blade were indicated. In both the model and the full scale rotor blade airload calculations a flat planar wake was assumed which is a good approximation at large advance ratios because the downwash is small in comparison to the free stream at large advance ratios. The large fluctuations in the measured airloads near the tip of the rotor blade on the advance side is predicted closely by the vortex lattice method. 17. Key Words (Suggested by Author(s)) 18. Distribution Statement Blade-vortex interaction Unclassified - Unlimited Vortex lattice method Unsteady aerodynamics ,:i _ ~ on. n ~ Rotor a irl oads .. 4'~ ~ ,~.; ., . ' .... ...,.. ·/~!;;l;:">·'\,"·" .~ ,.,:> •• ~ •• STAR Category: 01 19. Security Oassif. (of this report, 20. Security Classif. (of this' pariei .. ';, ,ir} No. of Pages 22. Price· Unclassified Unclassified 141 $4.75 • For sale bV the National Technical Information Service, Springfield, Virginia 22151 Page Intentionally Left Blank SUMMARY This study is concerned with the rotor blade-vortex interaction problem and the resulting impulsive air10ads which generate undesirable noise levels. First, a numerical lifting surface method is developed to predict unsteady aerodynamic forces induced on a finite aspect ratio rectangular wing by a straight, free vortex placed at an arbitrary angle in a subsonic incompressible free stream. Using a vortex lattice method unsteady air10ads on the wing are ob·tained by starting the system from rest. Load distributions are obtained which compare favorably with the results of planar lifting surface theory. The vortex lattice method is next extended to a single bladed rotor operating at high advance ratios and encountering a free vortex from a fixed wing upstream of the rotor. The predicted unsteady load distributions on the model rotor blade are generally in agreement with the experimental results. Finally full scale rotor flight cases having vortex induced loads near the tip of a rotor blade were calculated. Assumi·ng a flat planar wake the tip vortex from the preceding blade was placed at a specified height below the plane. of the rotor. The large fluctuations in the measured air10ads near the tip of the rotor blade on the advancing side are predicted closely by the vortex lattice method. Using conformal transformation methods an exact analysiS of the effects of thickness on the lift due to a two-dimensional wing-vortex interaction is presented. ii:i. Page intentionally left blank TABLE OF CONTENTS Page .:iUMMARY · . iii LIST OF TABLES. vi LIST OF FIGURES • vii NOMENCLATURE . xi I. INTRODUCTION'. 1 II. PREVIOUS INVESTIGATIONS · . 4 III. VORTEX LATTICE METHOD FOR WING-VORTEX INTERACTION PROBLEM • • • • • • • • • • • • • • • • • • • • • 10 3.1 Development of Unsteady Computational Model • 10 3.1.1 Analysis •••••••• 13 3.1.2 Results and Discussion ••• 21 IV. APPLICATION OF VORTEX LATTICE METHOD TO THE ROTOR­ VORTEX INTERACTION PROBLEM • • • • • • • • • • • • • 43 4.1 Model Rotor-Vortex Interaction Problem 43 4.1.1 The Model Rotor and Wake Configuration . · · · · 43 4.1. 2 Effect of Vortex Core. · 45 4.1. 3 Calculation Procedure. · · · . 47 4.1. 4 Results and Discussion · · · . 51 4.2 The Calculation of Helicopter Rotor Air1oads. · · · · 65 4.2.1 Calculation Procedure. · . · · 65 4.2.2 Results and Discussion · · · · · 74 V. CONCLUSIONS AND RECOMMENDATIONS . 104 BIBLIOGRAPHY ••• 108 APPENDIX A: Effect of Thickness on the Lift Due to Wing- Vortex Interaction • • 115 APPENDIX B: Radial Locations and Angle of Intersections Associated with Blade-Vortex Intersections • 122 v LIST OF TABLES Table Page 1 Model Rotor Operating Parameters • . • • • 52 2 Rotor Dimensions and Operating Conditions. • . 75 vi LIST OF FIGURES _ Figure Page 1 Biot-Savart Law. • • • •• 17 2 Unsteady Vortex Shedding • 19 3 Comparison of Numerical Solution to Wagner Function. 22 4 Development of Lift Following a Sudden Change in Angle of Attack. • • • • • • • • • •• 23 5 Chordwise Distribution of Circulation for C~ at Center Section. .. • • • • • • • • • • • .mlpt 25 6 Spanwise Distribution of Lift Coefficient. • 26 7 Wing-Vortex Interaction. 27 3 Effect of Time Step Size on Peak Wing Lift Coefficient. 29 9 Effect of Time Step Size on Peak Circulation • 30 10 Variation of Wing Lift Coefficient with Vortex Location for Different Time Increments •••• 31 11 Variation of Peak Circulation with Vortex Height 32 12 Variation of Peak Lift Coefficient with Vortex Height . 33 ., . 13 ,." Variation of Lift Coefficient with Vortex Location for Two-Dimensional (2-D) and Finite (3-D) Wings • 34 14 Variation of Lift Coefficient with Vortex Location for Different (foo/Vc) •••••••••••••••• 36 15 Variation of Lift Coefficient with Vortex Location for Different (h/c). • • • • • • • • • • • • • • • 37 16 Variation of Lift Coefficient with Vortex Location for hlc = 0.25 and hlc = -0.25 • • • • • • • • • • • •• 38 17 Effect of Steady and Quasi-Steady Assumptions on Lift Coefficient •••••••••••.•• 39 18 Variation of Lift Coefficient with Vortex Location for Different Aspect Ratios (A) • ••••••••• 40 19 Variation of Lift Coefficient with Vortex Location at Spanwise Station y/b = 0.95 •••••••••• 41 vii LIST OF FIGURES (continued) Figure Page 20 Rotary Wing Configuration for the Vortex Lattice Method • • • • • • • • • • . • • • 46 21 Blade Coordinates. 48 22 Example of Wake Configuration. • • 50 23 Typical Section Lift Coefficient Variation on Rotor . Blade. .. ..... 54 24a Variation of Maximum Section Lift Coefficient Difference (~C~) with Rotor Plane Position (h/c); Model Rotor, Z/R = 0.5 •••••••••••••• 55 24b Variation of Maximum Section Lift Coefficient Difference (~C~) with Rotor Plane Position (h/c); Model Rotor, Z/R =, 0.75 ••••••••••••• 56 25 Variation ~f Maximum Section Lift Coefficient Difference (~C5!) ~ith Rotor Plane Position (hI c): Model Rotor, Z/R - 0.5 ••••••••••..• 57 26a Spanwise Variation of Maximum Lift Difference (~L) Due to Vortex Interaction; Model Rotor, hlc = 0.5. 58 26b Spanwise Variation of Maximum Lift Difference (~L) Due to Vortex Interaction; Model Rotor, hlc = 0.25 59 27 Spanwise Variation of Maximum Lift Difference (~L) Due to Vortex Interaction; Model Rotor, hlc = 0.25 60 28 Spanwise Variation of MaXimum Lift Difference (~L) Due to Vortex Interaction; Model Rotor, hlc = 0.5 61 29a Spanwise Variation of Maximum Lift Coefficient Difference (~C~) Due to Vortex Interaction; ModeJ Rotor, hlc = 0.5 ••••••••••••••• 62 29b Spanwise Variation of Maximum Lift Coefficient Difference (~C~) due to Vortex Interaction; Model Rotor, hI c = 0.25. • . • • • • • • • • • • .• • 63 . 30 Spanwise Variation of Maximum Lift Coefficient Difference (~C~) Due to Vortex Interaction; Model Rotor, hlc = 0.5 • • • • • • • • • • • 64 31 Typical Variation of Section Lift on the Model Rotor Blade with Time (Azimuth) ••••••••••.••• 66 viji. LIST .OF FIGURES (continued) Figure Page 32 Relative Orientation of Reference Blade and Tip Vortex on the Advancing Side of Disk. • • • • • • • • • • 69 33 Velocity Components on a Blade in Forward Flight • 71 34 Comparison of Computed and Measured Section Lift; H-34 Helicopter Rotor, II == 0.2, r/R = 0.95. • • • • • • • •• 77 35 Comparison of Computed and Measured Section Lift; H-34 Helicopter Rotor, II :::: 0.2, r/R = 0.85. • • • • • • . • • 78 36 Comparison of Computed and Measured Section Lift; H-34 Helicopter Rotor, II = 0.2, r/R = 0.55. • • • • . • . • • 79 37 Comparison of Computed and Measured Section
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