Continuous Eulerian and Lagrangian Sensitivities for the Design of Airfoils in Laminar Flow

Abstract

´This paper presents a gradient-based optimal design procedure using the Continuous Lagrangian Sensitivity equation method (CLSEM) and a NURBS representation of the geometry. First, the CLSEM provides a more accurate evaluation of the derivatives of the objective functions than our previous approach using a Continuous Eulerian Sensitivity Equation Method (CESEM a.k.a. SEM or CSEM). Since this sensitivity formulation leads directly to the material derivatives of the dependent variables of the flow, the computation of the gradients with respect to shape parameters is simple and direct. Second, the geometric representation of airfoils based on NURBS reduces the number of design variables needed to accurately represent a wing section and ensures good smoothness properties. Furthermore the NURBS representation of airfoils is compatible with most CAD systems. The resulting optimal design procedure is described and then verified using the Method of Manufactured Solutions. It is applied to the design of airfoils in laminar flow at low Reynolds numbers.