1205.5516 (Qiqi Wang et al.)

The drag-adjoint field of a circular cylinder wake at Reynolds numbers
20, 100 and 500    [PDF]

Qiqi Wang, Junhui Gao

This paper analyzes the adjoint solution of the Navier-Stokes equation. We focus on flow across a circular cylinder at three Reynolds numbers, ReD = 20, 100 and 500. The objective function in the adjoint formulation is the drag on the cylinder. We use classical fluid mechanics approaches to analyze 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 ReD = 20, a viscous steady state flow, exhibits a downstream suction and an upstream jet, opposite of expected behavior of a flow field. The adjoint solution at ReD = 100, a periodic 2D unsteady flow, exhibits periodic, bean shaped circulation the near wake region. The adjoint solution at ReD = 500, a turbulent 3D unsteady flow, has complex dynamics created by the sheer layer in the near wake; the magnitude of the adjoint solution increases exponentially at the rate of the first Lyapunov exponent. The theoretical analysis and numerical results presented in this paper correlate well.

View original: http://arxiv.org/abs/1205.5516