An adjoint optimization method, based on the solution of an inverse flow problem, is proposed. Given a certain performance functional, it is necessary to find its extremum with respect to a flow variable distribution on the domain boundary, for example, pressure. The adjoint formulation delivers the functional gradient with respect to such a flow variable distribution, and a descent method can be used for optimization. The flow constraints are easily imposed in the parameterization of the distributed control, and therefore those problems with several strict constraints on the flow solution can be solved very efficiently. Conversely, the geometric constraints are imposed either by additional partial differential equations, or by penalization. By adequately constraining the geometric solution, the classical limitations of the inverse problem design can be overcome. Several examples pertaining to internal flows are given.

The equilibrium conditions of a point vortex in the separated flow past a locally deformed wall is studied in the framework of the two-dimensional potential flow. Equilibrium locations are represented as fixed points of the vortex Hamiltonian contour line map. Their pattern is ascribable to the Poincaré–Birkhoff fixed-point theorem. An ‘equilibrium manifold’, representing the generalization of the Föppl curve for circular cylinders, is defined for arbitrary bodies. The property $\partial\omega/\partial\skew3\tilde\psi\,{=}\,0$ holds on it, with $\skew3\tilde\psi$ being the stream function and $\omega$ the streamline slope of the pure potential flow.A ‘Kutta manifold’ is defined as the locus of vortices in flows that separate at a prescribed point (Kutta condition). The existence of standing vortices that satisfy the Kutta condition is discussed for symmetric bodies. On the basis of an asymptotic expansion of the equilibrium manifold, Kutta manifold and body geometry, it is shown that different classes of symmetric bodies exist which are ranked by the number of allowable standing vortices that satisfy the Kutta condition.

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Robust and flexible numerical methodologies for the imposition of boundary conditions are required to formulate well-posed problems. A boundary condition should be nonreflecting, to avoid spurious perturbations that can provocate unsteadiness or instabilities. The reflectiveness of various boundary conditions is analyzed in the context of the Godunov methods. A nonlinear, isentropic wave propagation model is used to investigate the reflection mechanism on the flowfield borders, and a parameter τ is defined to give a measure of the boundary reflectiveness. A new set of boundary conditions, in which τ = 0, that is, totally nonreflecting, is then proposed. The approach has been integrated in an aerodynamic design procedure using a distributed boundary control.