Counterflow around a cylinder

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 Kármán instability. The frequency of this mode is directly proportional to the strain rate defining the counterflow.

Figures (6)

• Figure 1: Schematic representation of the flow: a very long circular cylinder in an unconfined planar counterflow.
• Figure 2: Flow separation with recirculation bubbles of length $L_r$.
• Figure 3: The [l] dependence for: (a) recirculation bubble length, $L_r$, and maximum reversal speed, $U_{rev}$; (b) base pressure coefficient, $C^*_{p_b}$; (c) boundary layer separation angle, $\theta_{s}$, in degrees; and (d) coordinates of the $i$-th (i=1,2,3) vortex center, $[x_{1_c}^{(i)}, x_{2_c}^{(i)}]^T$.
• Figure 4: Topology of the recirculation zone for (a) $\Rey = 100$, (b) $\Rey = 1000$, (c) $\Rey = 2000$, and (d) $\Rey = 4000$, represented by streamlines and vorticity contours.
• Figure 5: Amplification rate ($\sigma$) and Strouhal number ($\St$) of the leading modes for each symmetry family (i.e., AA, AS, SA, and SS), as functions of the [l] ($\Rey$).