AN AUTOMATED APPROACH TO NACELLE PARAMETERIZATION USING INTUITIVE CLASS SHAPE TRANSFORMATION CURVES

A tool to create parametric aerodynamic shapes using intuitive design variables based on class shape transformation curves is presented. To enable this, a system has been developed which accepts arbitrary constraints and automatically derives the analytical expressions which describe the corresponding class shape transformation curves. Parametric geometry definitions for fan cowl and intake aero lines were developed using the generalized method. CFD analysis of the fan cowl shows that despite the simple geometry definition its performance characteristics are close to what would be expected of a finished design. The intake geometry was generated in a similar way and met the typical performance metrics for conventional intakes. This demonstrates the usefulness of the tool to quickly and robustly produce parametric aero lines with good aerodynamic properties, using relatively simple intuitive design variables. Nomenclature Roman Symbols a An element in the matrix of coefficients A Matrix of coefficients b An element in the matrix of constant terms B Matrix of constant terms C Class function c Chord length cD drag coefficient cD,hi nacelle drag coefficient related to highlight area CLmax Maximum lift configuration D f an Fan Diameter *Address all correspondence to this author. GTP 16 1227, Christie, Page 1 DC60 Total Pressure Distortion Parameter fmax Non dimensional position of maximum nacelle radius i Index of summation K Binomial coefficient k Order of derivative L2 norm The Euclidean norm Lintake Intake length M Mach number MISEN Isentropic Mach number N Upper bound on summation n Bernstein polynomial order N1 First exponent in the class function N2 Second exponent in the class function Ncon Number of constraints P0, f an f ace Fan face mean total pressure P0 Total pressure P60 Mean total pressure in the 60o sector with the lowest mean total pressure q Dynamic pressure Q f an Quasi non dimensional flow parameter at the nominal fan face R Radius of curvature r radial coordinate rhi highlight radius ri f initial forebody radius: radius of curvature at nacelle highlight rmax maximum nacelle radius rte nacelle trailing edge radius S Shape function x Abscissa X Matrix of solutions x axial coordinate y Ordinate Abbreviations AIP Aerodynamic interface plane AR Lip aspect ratio BP Bernstein Polynomial bp Bernstein polynomial weighting coefficient CFD Computational Fluid Dynamics CR Intake contraction ratio CRM Common Research Model CST Class Shape Transformation FPR Fan Pressure Ratio iCST Intuitive Class Shape Transformation IPR Intake Pressure Recovery le Leading edge LMFIT Non Linear Least Square Minimization and Curve Fitting for Python  Mass flow capture ratio PARSEC Parameterization method for airfoil sections TDC Top Dead Centre te Trailing edge TFN Through flow nacelle Greek Symbols a Angle of attack b Boattail angle dZte Radial offset between highlight and trailing edge k Curvature ps Non dimensional abscissa x Non dimensional ordinate GTP 16 1227, Christie, Page 2 INTRODUCTION External aerodynamic shapes can be represented in a number of ways. The most basic approach is to construct the shape from a point cloud. This method however requires a large number of design variables to guarantee a smooth and accurate shape  To limit the amount of design parameters, the aerodynamic shape can be defined by a parametric curve. The choice of parameterization method has a profound effect on the design space and is a key decision in the design process  Class shape transformation (CST) curves are known from wing profile design to create aerodynamically useful shapes.  This method is capable of producing a wide range of aerodynamic shapes including intake and fan cowl aero lines  in a universal way with a small number of design variables. The CST approach however lacks intuitiveness of the design parameters for geometrical and physical understanding of the constraints. This is required by a user in exploring the design space and in the application of optimization constraints. This drawback has been addressed for a specific set of constraints  The intuitiveness of the parameterization method for airfoil sections (PARSEC)  was combined with the highly flexible CST parameterization method. This intuitive CST (iCST) approach is a transformation of the CST parameterization method to a full set of intuitive parameters by a transformation matrix  The current work provides a generalization on this iCST approach. A system has been developed which analytically calculates the transformation matrix for a set of arbitrary constraints. This generalization of the iCST method has then been implemented in a Python based geometry tool. The usefulness of the generalized method for fan cowl and intake aero lines is then investigated. Representative fan cowl and intake aero lines are constructed based on the NASA common research model (CRM) [7] Nacelle drag characteristics are investigated through 2D axisymmetric Computational Fluid Dynamics (CFD) while the intake performance is assessed at high incidence with 3D CFD.  Firstly the analytical approach to the creation of a transformation matrix for intuitive class shape transformation parameterization will be discussed. The approach is illustrated with an example fan cowl parameterization. Details of the CFD methods and aerodynamic performance metrics used to assess fan cowl and intake parametric geometries are then presented.  represents a geometry by the product of a class function, C(ps) and a shape function, S(ps) The start and end points of all curves constructed with a class and shape function have equal ordinates. An additional term, (psxte) is included to modify the end point's ordinate  x(ps) S(ps)C(ps)+psxte x y c ps x c (1) The basic profile is defined by the class function. One set of class functions which describes basic external aerodynamic shapes through different exponent combinations takes the form  C1 N2 (ps) ps N1 [1 ps]N2 f or 0< ps< 1 (2) Equation 2 can be used to describe bi convex (C1.0 1.0) elliptic (C 0.5 0.5) and round nosed (C 1.0 0.5) airfoil shapes as well as other common external aerodynamic shapes such as a Sears Haack body (C0.75 0.75) or a cone (C 1.0 0. The class function is modified by the shape function. Bernstein polynomials (Eq. 3) are commonly employed as the shape function [3 5] BP(ps) N i=0 Ki,n psi (1 ps)n 1 Ki,n n! i!(n 1) (3) They are stable, robust and importantly form a partition of unity, i.e. the n+ 1 Bernstein polynomials of order n sum to one. The shape function can then be manipulated by the variation of Bernstein polynomial coefficients (Eq. 4) which in turn modifies the final profile (Figures 1 and 2) S(ps) N i=0 bpi *Ki,n psi (1 ps)n 1 (4) The mathematically smooth behavior of Bernstein polynomial shape functions and their inherent curvature continuity make GTP 16 1227, Christie, Page 3 0.0 0.2 0.4 0.6 0.8 1.0 0.96 0.98 1.00 1.02 1.04 1.06 S Weighted shape funct ion Unweighted shape funct ion Fig. 1: Perturbation of the shape function by variation of the Bernstein polynomial weighting coefficients 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Geom etry created with unweighted shape funct ion Geom etry created with weighted shape funct ion Fig. 2: Geometry formed using a unit shape function and a shape function perturbed by Bernstein polynomial weighting coefficients  this system ideal for aerodynamic optimization work. However the design relies on the manipulation of the Bernstein polynomial weighting coefficients which are not aerodynamically intuitive. To allow a design engineer to explore the design space and set up constraints for optimization process the design parameters need to be physically intuitive. The Bernstein polynomial coefficients can be analytically calculated from intuitive variables through the construction of a transformation matrix. This has been carried out for a specific set of constraints  However for each set of constraints a new transformation is required. Done manually this is a time consuming process. To rectify this a general method has been formulated which works with arbitrary constraints. This general method has then been implement in a Python based geometry generation tool. An automatic system to generate and solve these linear sets of equations has been developed. An arbitrary number of constraints (Ncon) can be specified. The order of the resultant CST curve is then Ncon 1. Position, gradient and derivatives up to the nth degree can be set. Compound constraints such as curvature, radius of curvature and their derivatives can also be imposed (Appendix A) For each constraint the only required inputs are its magnitude, x(k) and abscissa, ps. Then Eq. 5 is rearranged to isolate the Bernstein polynomial weighting coefficients. The function's coefficients and constant terms can then be analytically extracted. A Python symbolic math module  can then be employed to solve the class function and its derivatives. The shape function and its derivatives can be solved by using the fact that derivatives of the kth degree Bernstein GTP 16 1227, Christie, Page 4 polynomials are polynomials of degree k 1 and can be written as a linear combination of Bernstein polynomials (Eq. 6) x(k)(ps) N i=0 bpiKi,n psi (1 ps)n 1 C(ps)+psxte (k) (5) d dps BPi,n(ps) n[BPk 1,n 1(ps) BPk,n 1(ps) (6) A linear set of equations is then constructed with the extracted coefficients and constant terms (Eq. 7)