A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows

A continuous adjoint formulation for inverse design problems in external aerodynamics and turbomachinery is presented. The advantage of the proposed formulation is that the objective function gradient does not depend upon the variation of field geometrical quantities, such as metrics variations in the case of structured grids. The final expression for the objective function gradient includes only boundary integrals which can readily be calculated in both structured and unstructured grids; this is feasible in design problems where the objective function is either a boundary integral (pressure deviation along the solid walls) or a field integral (the entropy generation over the flow domain). The formulation governs inviscid and viscous flows; it takes into account the streamtube thickness variation terms in quasi-3D cascade designs or rotational terms in rotating blade design problems. The application of the method is illustrated through a number of design problems concerning isolated airfoils, a 3D duct, 2D, quasi-3D and 3D, stationary and rotating turbomachinery blades.