Spline-Based Airfoil Curvature Smoothing and Its Applications

Abstract

The performance of a transonic airfoil is directly related to the airfoil curvature profile and its smoothness. Whereas univariate data smoothing has been studied extensively, very little research has been conducted on curvature smoothing. Consequently, airfoil smoothing in design environments is largely based on heuristic methods, and there is an art to the generation of an unbiased smooth fit of the airfoil’s curvature profile by the modification of its geometry. In this paper, the sum of squares of the third derivative jumps is used as a curvature smoothness measure for the development of a spline-based airfoil smoothing method, called constrained fitting for airfoil curvature smoothing (CFACS). CFACS can take out dramatic curvature oscillations with extremely small geometry changes and smooth an airfoil segment without creating curvature oscillations near the endpoints. Visually, CFACS generates an unbiased smooth fit of the curvature profile. Examples demonstrating the utility of CFACS show how the smoothing can be tailored to promote desirable characteristics in performance trade studies. Nomenclature A = matrix representing smoothness measure c = chord length of airfoil cd = drag coefficient cl = lift coefficient c p = pressure coefficient f (t) = function of t for 0 ≤ t ≤ 1 f (r) (t) = r th derivative of f (t) f¯(t) = cubic spline interpolation of vector ¯ f ( 3) ¯ (t) = third derivative of f¯(t) M = freestream Mach number n = number of data points S(¯) = smoothness measure of vector ¯ t = column vector of t1, t2 ,..., tn ti = knot location of spline functions wi = positive weight in smoothness measure x = column vector of x coordinates of airfoil data xi, yi = coordinates of point on the plane y, ¯ = column vectors of y coordinates of airfoil data β, δ = parameters for choosing leading-edge segment � = percentage error tolerance for smoothing θ = control parameter (0 ≤ θ ≤ 1) for smoothing τi = positive weight for least-squares fitting �� = 2-norm of any vector