A Viscous Continuous Adjoint Approach for the Design of Rotating Engineering Applications

A viscous continuous adjoint formulation for optimal shape design is developed and applied. The arbitrary Lagrangian-Eulerian version of the unsteady, compressible Reynoldsaveraged Navier-Stokes (RANS) equations with a generic source term is considered, and from these governing ow equations, an adjoint formulation centered around nding surface sensitivities using dierential geometry is derived. This surface formulation provides the gradient information necessary for performing gradient-based aerodynamic shape optimization. To analyze the eectiveness of the methodology, two design cases in a rotating reference frame are considered. A two-dimensional test case consisting of a rotating airfoil at a low Reynolds number is studied. The shape of the airfoil is then optimized for drag minimization with a geometric constraint. In three-dimensions, the formulation is demonstrated using the well-known NREL Phase VI wind turbine geometry.

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