Loughborough University Institutional Repository Computer analysis of wing design for general aviation aircraft ThisitemwassubmittedtoLoughboroughUniversity’sInstitutionalRepository by the/an author. Additional Information: • AMastersDissertation,submittedinpartialfulfilmentoftherequirements of the award of Master of Science of Loughborough University. Metadata Record: https://dspace.lboro.ac.uk/2134/16174 Publisher: (cid:13)c Michael Papadakis Rights: This work is made available according to the conditions of the Cre- ative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/ Please cite the published version. -. - LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY LIBRARY . AUTHOR/FILING TITLE PAPA:i)A\(\S M - ---------------------/---------------------- -. - -------------------------------------- -------.....,... ACCESSION/COPY NO. ~--V--O-L~NO~- ~:lst-Utt------------------- '-3t) ...." -.''.! ( -7MAY 1 99 I I • -0106548 02 II\~\\\~\\\\\\\II\~~\\~\\\I\\\\\~\III\\I\\I\\~ . , • COMPUTER ANALYSIS OF WING DESIGN FOR GENERAL AVIATION AIRCRAFT BY MICHAEL PAPADAKIS A Thesis submitted in partial fulfilment of the requirements for the award of the degree of Master of Science of the Loughborough University of Technology January, 1981. Supervisors: F.G. MACCABEE, .. Eng.,A.F.R.Ae.S. M,~Sc.,D.C.Ae :.J~' . . t Department of Transport Technology .,' K.S. PEAT,'B.Sc.,Ph.D. " " Department of.Engineering Mathematics --" .. leughbor.ugh UnivenMoy ef TeeM";"" L'a>r..." ..,. M~St Cl IS. 10 ke.~-.,. I {) 6S l\-'i> 2- Ne, t SUMMARY The calculation of the two dimensional viscous incompressible flow about single and multielement aerofoil sections is considered. A panel method, based on vorticity and source distributions is used for the calculation of the potential flow. Once the velocity distribution is known, integral boundary layer methods are employed to predict the viscous effects. A wake model has also been developed for the calculation of the wake behind the aerofoil system. The solution is iterative. At the end of each iteration the velocities on the aerofoil are corrected for viscosity and wake effects; the wake position is also relaxed, before the next iteration starts. The mathematical model of the flow, together with the computer program written to test the model are described here in detail. The numerical results obtained using the computer program are found to be in good agreement with both experimental data and exact solutions. ,', . , , '. ACKNOWLEDGE~1ENTS The author wishes to acknowledge the co-operation and assistance he has received from many quarters; in particular he is indebted to his supervisors, Dr. Peat and Mr. Maccabee for their help and advice during the course of this research. The author also wishes to thank Mr. Tony Cross and Mr. David Butter, of British Aerospace, for their technical advice and Dr. Shanehchi who helped with the computer program. Finally for the work of typing thanks are due to Miss Judy Briers • .. 2 CONTENTS PAGE SUMMARY 1 2 ACKNOWLEDGEMENTS 3 CONTENTS 8 LIST OF FIGURES 11 LIST OF SYMBOLS INTRODUCTION 15 Section 1: MODELLING OF THE AIR FLOW - A GENERAL DISCUSSION 1.1 Introduction 19 1.2 Models to Describe the Air and Some of its Properties 19 1.3 Potential Flow 23. 1.4 Methods for Solving for the Flow Around an Aerofoil 26 1.4.1 Conformal Mapping 26 26 1.4.2 Singularities Section 2: THE HISTORY AND DEVELOPMENT OF TWO-DIMENSIONAL POTENTIAL FLOW METHODS 2.1 Introduction 28 2.2 The Problem of Calculating Wing Lift 28 2.3 The Early Stages 29 2.4 Development of the First Methods 30 33 2.5 Modern Methods 2.6 Concluding Remarks 38 Section 3: SURFACE SINGULARITY MODEL FOR THE CALCULATION OF THE POTENTIAL FLOW ABOUT THICK AEROFOIL SECTIONS 41 3.1 Introduction 3.2 Linear Vorticity Panel Method 42 3.2.1 Model Description 42 42 3.2.2 Equations 3.3 Symmetrical Linear Vortex and Constant Source Surface Singularity Model 3.3.1 Basic Considerations 43 45 3.3.2 Theoretical Basis 3.3.3 Numerical Solution Procedure 46 3.3.4 Velocity Equations 46 .. 3 PAGE Section 4: VISCOUS FLOW - A GENERAL DISCUSSION 4.1 Introduction 50 4.2 Genesis of the Boundary Layer Concept 50 4.3 Viscous Flow Model 52 4.4 Viscous Flow Representation 54 4.5 Aerofoil Surface Velocity Equations 56 Section 5: LAMINAR BOUNDARY LAYER 5.1 Introduction 59 5.2 Thwaites Method 59 5.3 CurIe's Method 61 5.3.1 Equations 62 5.3.2 Calculation Procedure 62 5.3.3 Laminar Separation 63 Section 6: BOUNDARY LAYER TRANSITION 6.1 Introduction 67 6.2 Crab tree Criterion 67 6.3 Michel's Method 68 6.4 Granville's Method 68 6.4.1 Instability Prediction 69 6.4.2 Transition Prediction 69 Section 7: SHORT BUBBLE ANALYSIS 7.1 Introduction 72 7.2 Short Bubble Model of Shear Layer 73 7.2.1 Laminar Part of Shear Layer 74 7.2.2 Turbulent Part of Shear Layer 75 7.2.3 Bubble Length ~t Bursting 76 7.3 Method of Solution 77 Section 8: TURBULENT BOUNDARY LAYER 8.1 Introduction 80 8.2 Head's Entrainment Method 80 81 8.2.1 Equations 8.2.2 Method of Calculation 81 8.3 The Method of Head and Patel 82 83 8.3.1 Equations 83 8.3.2 Method of Calculation 84 8.4 Initial Values 8.5 Turbulen.t Boundary Layer Separation 8.5.1 Head's Criterion 85 8.5.2 Epplers Method 85 86 8.6 toncluding Remarks 4 PAGE Section 9: WAKE ANALYSIS 9'.1 Introduction 92 9.2 Potential Flow Wake Model 92 9.3 Wake Singularity Sheet 93 9.4 Velocity Equations 94 9.5 Wake Viscous Flow Analysis 95 9.5.1 Equations 95 9.5.2 Initial Conditions ' 97 9.5.3 Calculation Procedure 97 9.6 Calculation of Wake Singularities 97 9.7 Wake Relaxation 98 9.8 End of Potential Core 100 9.9 Aerofoil Velocities 101 Section 10: CONFLUENT BOUNDARY LAYER 10.1 Introduction 107 10.2 General Considerations 107 10.2.1 Main Assumptions 108 10.3 Model Description 108 10.4 Equations for the Calculation of Main Region 1 and 2 110 10.4.1 Coles Velocity Profile 110 10.4.2 Main Region 1 111 10.4.3 Main Region 2 115 10.5 Initial Conditions 117 10.5.1 Main Region 1 is Preceded by a Core Region 117 10.5.2 Main Region 2 Not Preceded by Core 118 10.5.3 Main Region 2 119 10.6 Calculation Procedure 119 10.7 Boundary Layer Parameters ~' Conf1ue~t 10.7.1 Thickness 0 119 4 10.7.2 Displacement: Thickness 0 * 120 10.7.3 Momentum Thickness e 120 10.7.4 Skin Friction 120 10.7.5 Separation 121 10.8 Further Comments 121 Section 11: SEPARATED FLOW MODEL 11.1 Introduction 125 11.2 Flow Separation 125 11.3 Separation Flow Model, 127 11.3.1 Basic Considerations and Assumptions , 127 11.3.2 Approximations for the Free Shear Layer 129 11.3.3 . Theoretical Basis 130 11.3.4 Method of Solution 130 11.4 .. Concluding Remarks 132 5