Project Department of Automotive and Aeronautical Engineering Mach number, relative thickness, sweep and lift coefficient of the wing - An empirical investigation of parameters and equations Author: Simona Ciornei Examiner: Prof. Dr.-Ing. Dieter Scholz, MSME Delivered: 31.05.2005 2 Abstract 12 equations were investigated to calculate the relative thickness t/c of the wing of an aircraft. The calculated relative thickness was taken as the average relative thickness of the wing. The data obtained from these 12 equations was checked against the given average relative thickness of 29 carefully selected aircraft, spanning a space of the parameters Mach number, lift coefficient, sweep, and type of airfoil. Some equations selected are empirical in their nature (partly based on aerodynamic derivation) other equations are purely statistical. Whenever equations had free parameters, these were optimized against the aircraft data. The best equation turned out to be an equation based on nonlinear regression. It achieved a Standard Error of Estimate of only 0.75 % for the average relative thickness of the wing. Torenbeek’s equation will probably be preferred by those that like to see an equation that is based on aerodynamic considerations. It achieved a Standard Error of Estimate of 0.80 % when all its free parameters were considered for optimization. The worst equation produced an Standard Error of Estimate of 8 %. For an airfoil with 10 % relative thickness this would give an unacceptable 10 % +/- 8 % band of values for t/c which renders equations like this quite useless. 1 FACHBEREICH FAHRZEUGTECHNIK UND FLUGZEUGBAU Mach number, relative thickness, sweep and lift coefficient of the wing - an empirical investigation of parameters and equations Aufgabenstellung zum Projekt gemäß Prüfungsordnung Background In aircraft design, the wing parameters "relative thickness" and "sweep" follow from a demand for a certain cruise Mach number at low wave drag. In addition, the cruise lift coefficient and the type of airfoil have an influence on the aerodynamics of the wing. If there is a demand for a higher cruise Mach number during aircraft design, the sweep has to be increased or the rela- tive thickness has to be decreased. The transonic flow around a wing can not be described with simple equations. For this reason, the relationship between the parameters as given above will be based in preliminary aircraft design on statistics of known aircrafts. Task Equations based on statistical data relating Mach number, relative thickness, sweep and lift coefficient of the wing have to be investigated, checked and improved for their suitability in preliminary aircraft design. The project's task includes these subtasks: • Introduction to transonic flow around wings. • Literature search for equations dealing with the relationship of named parameters. • Theoretical substantiation of the empirical equations as far as possible. • Investigation of aircraft parameters for sample calculations with equations form the litera- ture. • Comparison of equations based on sample calculations. Selection of the most suitable equation. • Adaptation of this equation to further improve the accuracy based on given aircraft pa- rameters. The report has to be written according to German DIN standards on report writing! 4 Table of Contents page List of Figures ........................................................................................................................... 8 List of Tables .......................................................................................................................... 10 List of Symbols ....................................................................................................................... 11 List of Key Words and Definitions ......................................................................................... 13 1 Introduction ........................................................................................................ 15 1.1 Motivation ............................................................................................................ 15 1.2 Definitions ............................................................................................................ 15 1.3 Task ...................................................................................................................... 18 1.4 Literature .............................................................................................................. 19 1.5 Structure of Work ................................................................................................ 19 2 Transonic Flow ................................................................................................... 21 2.1 Transonic Flow Phenomena ................................................................................. 21 2.2 Compressibility Corrections ................................................................................ 23 2.3 Critical Mach Number and Critical Pressure Coefficient .................................... 24 2.4 Drag –Divergence Mach Number ........................................................................ 28 2.5 Development of the Supercritical Airfoil ............................................................. 31 2.6. Swept Wings ........................................................................................................ 34 2.7 Relative Thickness ............................................................................................... 37 3 Equations for the Calculation of Relative Thickness ..................................... 38 3.1 Equation based on Torenbeek .............................................................................. 38 3.2 Equations from Aerodynamic Similarity based on Anderson .............................. 40 3.3 Equation based on Shevell ................................................................................... 41 3.4 Equation based on Kroo ....................................................................................... 44 3.5 Equation from Howe ............................................................................................ 47 3.6 Equation from Jenkinson ..................................................................................... 48 3.7 Equation from Weisshaar ..................................................................................... 49 3.8 Equation based on Böttger ................................................................................... 50 3.9 Equation based on Raymer ................................................................................... 54 3.10 Equation based on Linear Regression .................................................................. 59 3.11 Equation based on Nonlinear Regression ............................................................ 60 3.12 Substantiation of the Equation from Torenbeek .................................................. 61 4 Investigation Comparison and Adaptation of Equations ............................... 65 4.1 Input from Aircraft Data ...................................................................................... 65 4.2 Calculation, Optimization and Results ................................................................ 66 5 5 Conclusions ......................................................................................................... 70 6 Recommendations ............................................................................................... 71 List of References................................................................................................................... 72 Appendix A Three-View Drawings .................................................................................... 77 Appendix B Investigation of Aircraft Parameters from Different Sources ................. 107 Appendix C Summary of Aircraft Parameters .............................................................. 137 Appendix D Calculation of Relative Thickness / Optimization of Equations ............. 142 Appendix E Schaufele’s Method .......................................................................................151 6 List of Figures Figure 1.1 Definition of sweep angles on a tapered inner and outer wing .........................17 Figure 2.1 Physical mechanism of drag divergence ...........................................................22 Figure 2.2 Variation of (a) lift coefficient and (b) drag coefficient versus Mach number with angle of attack as a parameter for an NACA 2315 airfoil............22 Figure 2.3 Several compressibility corrections compared with experimental results for an NACA 4412 airfoil at an angle of attack α = 1°53’ ...............................24 Figure 2.4 Illustration of critical Mach number..................................................................25 Figure 2.5 Illustration of critical pressure coefficient.........................................................26 Figure 2.6 Critical pressure coefficient and critical Mach numbers for airfoils of different thickness.........................................................................................26 Figure 2.7 Sketch of the variation of profile drag coefficient with freestream Mach number, illustrating the critical and drag-divergence Mach number and showing the large drag rise near Mach 1.............................29 Figure 2.8 Definition of critical Mach number...................................................................30 Figure 2.9 Peaky