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

TL;DR

This study investigates the adjoint solutions of the Navier-Stokes equations around a circular cylinder at various Reynolds numbers, revealing flow-dependent behaviors and complex dynamics relevant for flow control and sensitivity analysis.

Contribution

The paper provides a detailed analytical and numerical analysis of the adjoint field for different flow regimes around a cylinder, linking theoretical insights with computational results.

Findings

Adjoint flow at Re_D=20 shows downstream suction and upstream jet.

At Re_D=100, the adjoint exhibits periodic circulation patterns.

Re_D=500 adjoint displays complex, turbulent-like dynamics.

Abstract

This paper analyzes 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 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 Re_D=20, a viscous steady state flow, exhibits a downstream suction and an upstream jet, opposite of the expected behavior of a flow field. The adjoint solution at Re_D=100, a periodic 2D unsteady flow, exhibits periodic, bean shaped circulation in the near wake region. The adjoint solution at Re_D=500, a turbulent 3D 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.