Airfoil Parameterization using An Orthogonal Class Shape Transformation

Abstract

Reduced-order (or data-driven) modeling for predicting airfoil, blade, and wing aerodynamic loads has been studied extensively due to the relatively fast prediction time and low computational cost. The process starts with producing vast amount of training data (2D or 3D CFD simulations) and then training the models to represent the behavior of the data. Typically, airfoil coordinates are parameterized to yield only a few inputs that can be used in the training process for the reduced-order model. Various airfoil parameterization methods based on airfoil geometric quantities, analytical functions, and dimensional reduction were considered. In this paper, a dimensional reduction method based on Class Shape Transformation (CST) methodology is used to parameterize the airfoils. Previous researchers used shape functions that are not orthogonal to the class function (but some are orthogonal to their respective weighting functions). A new set of orthogonal shape functions is derived based on a class function that is specifically used to represent airfoils for CST methodology. The CST with orthogonal shape functions (CSTO) is then used to parameterized two sets of airfoil databases (SC1095 variants with 625 airfoils, UIUC airfoil database with 1621 airfoils). Some 4% of the airfoils in UIUC airfoil database are screened out due to various reasons including airfoil closeness, small chord length, small thickness, and poor point density. The rest of the airfoils are parameterized using the CSTO methodology. The parameterization error (squared L2 norm) for CST using traditional shape functions and the CSTO is compared for various NACA airfoils, and the CSTO shape functions consistently produce lowest fitting error. The effects of CSTO polynomial order and airfoil coordinate quality are studied, and the CSTO representation seems to converge as higher order terms are included. The numerical behaviors of CSTO parameters (for screened airfoils in UIUC database) are examined to be: - i) normally distributed, ii) asymptotic to zero [for higher order terms]. Overall CSTO with 8th order polynomials can successfully parameterize most airfoils in UIUC database (70% of the airfoils have fitting error of less than 1e-5, the mean fitting error of 2e-6, and 1% of the airfoils has fitting error greater than 1e-4). The CSTO parameters are used to classify groups of airfoils in the SC1095 database using k-mean clustering algorithm, and the results show that the k-mean algorithm is very good at grouping the airfoils into distinct groups even in the presence of noise. An airfoil identification procedure based on CSTO parameters is developed and the identification methodology can successfully identify airfoil when noise is less than 0.001c.

Topics & Concepts