Airfoil Design & Analysis Using Conformal Transformations

Introduction Designing and analyzing airfoils is crucial in aerospace applications. This report explores a simple but efficient method using Joukowski and Karman-Trefftz transformations to generate airfoil geometry. Using the potential flow theory and assuming inviscid incompressible flow, the method determines the surface velocities and pressure coefficient distributions, followed by an analysis of the airfoils’ overall Cl (lift coefficient). An investigation into the lift coefficient vs angle of attack performance of the designed airfoil is directly compared to the NACA 0012 and 0015 for model verification. After the validation of performance results, a Karman-Trefftz transformation (variation of Joukowski theorem) is utilized to enhance the flow model accuracy and manufacturability of the airfoil. Purpose The MATLAB model constructs and provides a quick analysis of 2D Joukowski-style airfoils. Though its current application solely applies to incompressible, inviscid flow, further enhancement of viscid flow analysis for both Joukoski and Karman-Trefftz airfoils will be possible. It is designed for early-stage airfoil development in non-critical applications, such as drones, offering a cost-effective alternative to computationally intensive CFD (computational fluid dynamics) simulations. Due to the simplistic nature of the model, the overall lift performance is believed to contain a minor degree of inaccuracy compared to the NACA airfoils under investigation. The model's objective is to provide an alternative to full-scale airfoil design processes for smaller-scale applications. Method Joukowski airfoils are designed using a conformal mapping transformation that transforms two circles about an arbitrary one-dimensional plane, z, into an airfoil about a related two-dimensional plane, . The airfoil depends on the one-dimensional plane’s geometric x and y central coordinates; obtained through the conformal transformation. Following its creation, potential flow analysis is used to model the stagnation and resulting upper and lower streamlines on the Joukowski airfoil. The analysis assumes steady, inviscid, incompressible, and two-dimensional flow. Aerodynamic analysis of the airfoils is conducted using coefficients of pressure formulas and coefficients of lift formulas. The pressure distribution is created by relating the pressure coefficient to the dimensionless chord position, defined as the chord position relative to the chord length. The lift coefficient is achieved through an integration analysis of the pressure coefficient curves of the upper and lower surfaces. To ensure manufacturability of the airfoil shape, the Karmen-Trefftz transformation is applied to adjust the Joukowski’s trailing edge through the manipulation of set parameters that control the thickness, leading-edge curvature, trailing edge curvature, and camber. Throughout the approach, a circulation factor is used to model varying airspeeds acting on the airfoil, allowing the airfoil’s lift performance to match the NACA airfoil performance without needing to alter the airfoil’s geometry or angle of attack. Model The NACA 4-digit series 0012 and 0015 airfoils are defined by two key features: geometric shape and performance characteristics. These features will be used to compare the performance characteristics of airfoils designed within a MATLAB program using Joukowski Conformal Transformations and a potential flow analysis approach. A “circulation” factor will represent the effects of varying airspeeds to simulate the airflow around the airfoil in MATLAB. This approach is predicted to carry a minor degree of inaccuracy regarding airfoil performance and is dissimilar to the CFD software. The geometry of the 0012 and 0015 foils are symmetric such that the first two digits are 0%, indicating that there is no camber and this position is in the middle of the airfoil. The last two digits are 12 and 15, where the maximum thicknesses of each airfoil are 12% and 15% of the cord length, respectively. The airfoils, although symmetric, have different thicknesses. NACA 0012 has zero lift at zero angles of attack, and the NACA 0015 is thicker than the NACA 0012; its lift is greater at similar angles of attack than the NACA 0012 airfoil. Thus the NACA 0015 is capable of maximum lift before stalling, which is suitable for aircraft wings that require low speed and higher lift. The NACA 0012 has a lower lift coefficient and increases at a slower rate than the NACA 0015 and thicker airfoils. The model can accurately produce lift coefficient values for angles of attack between 0 and 8 degrees. The model does not account for flow separation and non-linear effects; therefore, it is less accurate when analyzing angles of attack above 8 degrees. The calculated coefficient of lift values can be compared to the existing NACA 0012 and NACA 0015 data using graphs relating the coefficient of lift with respect to the angle of attack. Analysis The accuracy of our geometrical model can be confirmed by comparing the plotted Cl with respect to the angle of attack. Since it is a non-critical system, using multiple iterations we expect an accuracy of at least 90% compared to the plots for the NACA 0012 and 0015 found within the NACA database. Some discrepancies that may cause errors are that different design methods are used to construct the airfoil. The NACA airfoils are designed using analytical equations that describe the airfoils camber, mean line, and thickness distribution along the chord line. The MATLAB model uses the Joukowski and Karmen-Trefft transformations to create a similar airfoil. Expected Findings The model is expected to produce airfoil designs with geometry, streamlines, and Kutta conditions using Joukowski and Karman-Trefftz transformations that are comparable to the NACA 0012 and NACA 0015. The model is also expected to produce pressure coefficient vs chord length graphs for the values along the upper and lower surfaces of the airfoil following the necessary calculations for the surface airflow velocities. The pressure coefficients at the trailing edge are expected to spike for the Joukowski airfoil due to the infinitely small finish produced by Joukowski transformation and decrease in a conformal manner for Karman-Trefftz adaption due to the cusped trailing edge. These performance characteristics are predicted to be within 90% of the Cl vs Alpha graph database values of NACA, as the geometry produced by the transformation models is aimed to represent the NACA airfoils closely. Therefore, the model provides a simple design method, efficient airfoil shape, and performance analysis with reasonable accuracy.