An adjoint system of the Euler equations of gas dynamics is derived in order to solve aerodynamic shape optimization problems with gradient-based methods. The derivation is based on the fully discrete flow model and involves differentiation and transposition of the system of equations obtained by an unstructured and node-centered finite-volume discretization. Solving the adjoint equations allows an efficient calculation of gradients, also when the subject of optimization is described by hundreds or thousands of design parameters.
Such a fine geometry description may cause wavy or otherwise irregular designs during the optimization process. Using the one-to-one mapping defined by a Poisson problem is a known technique that produces smooth design updates while keeping a fine resolution of the geometry. This technique is extended here to combine the smoothing effect with constraints on the geometry, by defining the design updates as solutions of a quadratic programming problem associated with the Poisson problem.
These methods are applied to airfoil shape optimization for reduction of the wave drag, that is, the drag caused by gas dynamic effects that occur close to the speed of sound. A second application concerns airfoil design optimization to delay the laminar-to-turbulent transition point in the boundary layer in order to reduce the drag. The latter application has been performed by the author with collaborators, also using gradient-based optimization. Here, the growth of convectively unstable disturbances are modeled by successively solving the Euler equations, the boundary layer equations, and the parabolized stability equations.
Note: In the first printed version, many references to the bibliography are off by one. The online version does not have this error
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