Old Dominion University ODU Digital Commons Mechanical & Aerospace Engineering Theses & Mechanical & Aerospace Engineering Dissertations Summer 1984 Studies on the Interference of Wings and Propeller Slipstreams Ramadas K. Prabhu Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mae_etds Part of the Aeronautical Vehicles Commons, and the Mechanical Engineering Commons Recommended Citation Prabhu, Ramadas K.. "Studies on the Interference of Wings and Propeller Slipstreams" (1984). Doctor of Philosophy (PhD), dissertation, Mechanical & Aerospace Engineering, Old Dominion University, DOI: 10.25777/ezag-d204 https://digitalcommons.odu.edu/mae_etds/262 This Dissertation is brought to you for free and open access by the Mechanical & Aerospace Engineering at ODU Digital Commons. It has been accepted for inclusion in Mechanical & Aerospace Engineering Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. STUDIES ON THE INTERFERENCE OF WINGS AND PROPELLER SLIPSTREAMS by Ramadas K. Prabhu M.E. (Aeronautical Engineering) 1963 Indian Institute of Science, Bangalore, India A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirement of the Degree of DOCTOR OF PHILOSOPHY MECHANICAL ENGINEERING OLD DOMINION UNIVERSITY August, 1984 Approved by: S. N. Tiwari (Director) 0. A. Kandil C. H. Liu (Co-Director) C. H. Cooke Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT STUDIES ON THE INTERFERENCE OF WINGS AND PROPELLER SLIPSTREAMS Old Dominion U niversity, 1984 Director: Dr. Surendra N. Tiwari with Dr. Chen-Huei Liu The small disturbance potential flow theory is applied to determine the l i f t of an a ir fo il in a nonuniform p arallel stream. The given stream is replaced by an equivalent stream with a certain number of velocity discontinuities, and the influence of these discontinuities is obtained by the method of images. Next, this method is extended to the problem of an a ir fo il in a nonuniform stream of smooth velocity profile. This model allows perturbation velocity potential in a rotational undisturbed stream. A comparison of these results with numerical solutions of Euler equations indicates that, although approximate, the present method provides useful information about the interaction problem while avoiding the need to solve the Euler equations. The assumptions of the classical lif t in g lin e theory applied to the wing-slipstream interaction problem are scrutinized. One of the assumptions (uniform velocity in the slipstream) of the classical theory is dropped, and the governing equations are derived for the spanwise lift distribution on a wing in a single axisymmetric slipstream. Spanwise l i f t and induced drag distributio ns are obtained for two typical cases, and the effects of nonuniformity in the slipstream Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. velocity p ro file are examined. The method of matched asymptotic expansions is applied to the problem of a large aspect ra tio swept wing in the slipstream of m ultiple overlapping propellers. The flow is assumed to be steady, inviscid and incompressible. It is also assumed that the height of the slipstream is of the order of the wing chord, and its spanwise extent is of the order of the wing span. Three d iffe re n t flow regions are id e n tifie d by employing d iffe re n t stretching transformations, and asymptotic expansions are introduced using the chord-to-span ra tio as the small expansion parameter. The d etails of the nonuniform flow in the slipstream enter into the wing-sectional analysis. In the outer lim it, the wing shrinks to a swept lif tin g lin e , and the slipstream reduces to a thin sheet of jet carrying the momentum gain from the propeller. The curvature of this jet sheet results in a pressure difference which is represented by a vortex sheet. The governing equations are solved by discretization. Comparison of the present results with the experimental data as well as other numerical solutions showed generally good agreement. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to his advisors Dr. S. N. Tiwari and Dr. C. H. Liu for proposing this research topic, and guidance during the course of the study. The author is indebted to R. J. Margason, Head of NASA Langley Analytical Methods Branch for his constant encouragement during this endeavor. Also, the author wishes to thank Drs. C. H. Cooke, 0. A. Kandil and J. M. Kuhlman for being on the thesis committee. The author wishes to thank Ms. Diane Riddick for her timely typing of the manuscript. F in a lly , the author extends gratitude to his wife Sumithra for her constant encouragement and endurance during the course of his studies. The work reported here was supported by NASA Langley Research Center/Old Dominion University Research Foundation Cooperative Agreement NCC1-65, and was monitored by Dr. C. H. Liu. i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page ACKNOWLEDGEMENT ......................................................................................................... i LIST OF FIGURES ......................................................................................................... iv LIST OF SYMBOLS ......................................................................................................... vi Chapter 1. INTRODUCTION ..................................................................................................... 1 2. AIRFOIL IN NONUNIFORM PARALLEL STREAMS ................................................ 10 2.1 Fundamental Basis for the Analysis .............................................. 11 2.2 A irfo il in a Jet of Finite Width .................................................. 15 2.3 A irfo il in the Middle of Five P arallel Streams ........................ 19 2.4 Airfoil in a Stream of Smooth Velocity Profile ........................ 24 2.5 Results and Discussion ....................................................................... 28 3. A MODIFIED LIFTING-LINE THEORY FOR WING-SLIPSTREAM INTERFERENCE ..................................................................................................... 38 3.1 Governing Equations for a Single Slipstream ............................ 40 3.2 Method of Solution ............................................................................... 49 3.3 Results and Discussion ....................................................................... 52 4. AN ASYMPTOTIC THEORY FOR THE INTERFERENCE OF SWEPT WINGS AND MULTIPLE SLIPSTREAMS ..................................................................................... 59 4.1 Mathematical Formulation ................................................................... 59 4.2 The Inner Region ................................................................................... 64 4.3 The Third Region ................................................................................... 69 4.4 The Outer Region ................................................................................... 71 i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter Page 4.5 Method of Solution ............................................................................... 76 4.6 Results and Discussion ...................................................................... 80 5. CONCLUDING REMARKS ......................................................................................... 94 REFERENCES .................................................................................................................... 98 i i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure Page 2.1 A body near a surface of velocity discontinuity .................... 12 2.2 Image system for an airfoil near a surface of velocity discontinuity ....................................................................... 14 2.3 Image system for an airfoil in the middle of three p a ra lle l streams ................................................................................... 16 2.4 Image system for an airfoil in the middle of five p a ra lle l streams ................................................................................... 20 2.5 Image system for an a ir fo il in a stream of stepped velocity profile .......... 25 2.6 L if t curve slope of a f l a t plate in a nonuniform flow ............................................................................................................ 30 2.7 L if t curve slope of a f l a t plate in a nonuniform flow ............................................................................................................ 31 2.8 Chordwise pressure distribution on a flat plate in a nonuniform flow ..................................................................................... 33 2.9 Lift curve slope of a flat plate in a nonuniform flow ............................................................................................................ 34 3.1 The schematic of the classical analysis .................................... 39 3.2 The schematic of the present