Journal
JOURNAL OF FLUID MECHANICS
Volume 730, Issue -, Pages 145-161Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/jfm.2013.323
Keywords
chaos; variational methods; wakes
Categories
Funding
- DOE [FA9550-12-C-0065]
- NASA Fundamental Aeronautics Program
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This paper analyses the adjoint solution of the Navier-Stokes equation. We focus 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 on the cylinder. We use classical fluid mechanics approaches to analyse the adjoint solution, which is a vector field similar to a flow field. Production and dissipation of kinetic energy of the adjoint field is discussed. We also derive the evolution of circulation of the adjoint field along a closed material contour. These analytical results are used to explain three numerical solutions of the adjoint equations presented in this paper. The adjoint solution at Re-D = 20, a viscous steady state flow, exhibits a downstream suction and an upstream jet, the opposite of the expected behaviour of a flow field. The adjoint solution at Re-D = 100, a periodic two-dimensional unsteady flow, exhibits periodic, bean-shaped circulation in the near-wake region. The adjoint solution at Re-D = 500, a turbulent three-dimensional unsteady flow, has complex dynamics created by the shear layer in the near wake. The magnitude of the adjoint solution increases exponentially at the rate of the first Lyapunov exponent. These numerical results correlate well with the theoretical analysis presented in this paper.
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