Counterflow around a cylinder

TL;DR

This study investigates the flow behavior around a cylinder in a counterflow, revealing flow separation, recirculation patterns, and wake instability through numerical and stability analysis.

Contribution

It provides new insights into flow separation, recirculation development, and wake instability thresholds in counterflow conditions around a cylinder.

Findings

Flow separates at Re ≈ 16.86, forming symmetric recirculation zones.

Multiple recirculation centers emerge at higher Re, similar to Moffatt eddies.

Wake instability occurs at Re ≈ 4146, leading to oscillatory wake meandering.

Abstract

The incompressible flow around a circular cylinder, positioned at the center of an unconfined planar counterflow, is studied by means of numerical solutions of the conservation equations and linear stability analysis. The flow is completely defined by the Reynolds number ($\Rey$) -- based on the cylinder radius, the strain rate defining the counterflow, and the kinematic viscosity. For very low values of $\Rey$, the flow is steady, two-dimensional, and fully attached to the cylinder wall. Increasing $\Rey$ above $\Rey_s \approx 16.86$, the flow separates, giving rise to two symmetric, counter-rotating recirculation regions on each side of the cylinder. Further increasing $\Rey$ leads to a progressive enlargement of the recirculation regions and the appearance of multiple recirculation centers, akin to Moffatt eddies. However, the convective acceleration imposed by the counterflow limits their size. An oscillatory mode becomes linearly unstable for $\Rey_{c} \approx 4146$. This mode gives rise to a sinuous meandering of the wake flow, on each side of the cylinder, being analogous to the well-known von K\'arm\'an instability. The frequency of this mode is directly proportional to the strain rate defining the counterflow.