This paper analyzes the adjoint solution of the Navier-Stokes equation. Wefocus on flow across a circular cylinder at three Reynolds numbers, Re_D=20,100 and 500. The quantity of interest in the adjoint formulation is the drag onthe cylinder. We use classical fluid mechanics approaches to analyze theadjoint solution, which is a vector field similar to a flow field. Productionand dissipation of kinetic energy of the adjoint field is discussed. We alsoderive the evolution of circulation of the adjoint field along a closedmaterial contour. These analytical results are used to explain three numericalsolutions of the adjoint equations presented in this paper. The adjointsolution at Re_D=20, a viscous steady state flow, exhibits a downstream suctionand an upstream jet, opposite of the expected behavior of a flow field. Theadjoint solution at Re_D=100, a periodic 2D unsteady flow, exhibits periodic,bean shaped circulation in the near wake region. The adjoint solution atRe_D=500, a turbulent 3D unsteady flow, has complex dynamics created by theshear layer in the near wake. The magnitude of the adjoint solution increasesexponentially at the rate of the first Lyapunov exponent. These numericalresults correlate well with the theoretical analysis presented in this paper.
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