Optimal Shape Design for Open Rotor Blades

Abstract

A continuous adjoint formulation for optimal shape design of rotating surfaces, including open rotor blades, is developed, analyzed, and applied. The compressible Euler equations are expressed in a rotating reference frame, and from these governing flow equations, an adjoint formulation centered around finding surface sensitivities using di↵erential geometry is derived. The surface formulation provides the gradient information necessary for performing gradient-based aerodynamic shape optimization. A two-dimensional test case consisting of a rotating airfoil is used to verify the accuracy of the gradient information obtained via the adjoint method against finite di↵erencing, and a gradient accuracy study is also performed. The shape of the airfoil is then optimized for drag minimization in the presence of transonic shocks. In three-dimensions, the formulation is verified against finite di↵erencing for a classic, two-bladed rotor, which is then redesigned for minimum inviscid torque using a Free-Form Deformation approach to geometry parameterization. Optimal shape design for open rotor blades is presented as a final application of the new continuous adjoint formulation.