Square cylinder in the interface of two different-velocity streams

TL;DR

This study examines how a square cylinder in a two-stream flow transitions from vortex shedding to chaos as the velocity ratio increases, revealing new instability modes and complex spatiotemporal behaviors.

Contribution

It provides new insights into the flow instabilities and transition to chaos for a square cylinder in shear flows with varying velocity ratios, including the identification of mode-C instability and spatio-temporal chaos.

Findings

Vortex shedding onset is delayed at higher velocity ratios.

A mode-C instability triggers three-dimensional flow at R≈3.1.

Flow becomes spatio-temporally chaotic at R≥4.

Abstract

We investigate the incompressible flow past a square cylinder immersed in the wake of an upstream nearby splitter plate separating two streams of different velocity. The bottom stream Reynolds number, based on the square side, $Re_B=56$ is kept constant while the top-to-bottom Reynolds numbers ratio $R\equiv Re_T/Re_B$ is increased in the range $R\in[1,6.5]$, corresponding to a coupled variation of the bulk Reynolds number $Re\equiv(Re_T+Re_B)/2\in[56,210]$ and an {\it equivalent} nondimensional shear parameter $K\equiv2(R-1)/(R+1)\in[0,1.4667]$. The onset of vortex-shedding, at $R=2.1\pm0.1$ (corresponding to $Re=86.8\pm2.8$, $K=0.71\pm0.04$), is pushed to higher $Re$ as compared to the square cylinder in the classic configuration. The advent of three-dimensionality is triggered by a mode-C-type instability at $R\simeq3.1$ ($Re\simeq115$, $K\simeq1.02$) with wavelength $\lambda_z\simeq2.4$, much as reported for open circular rings and square cylinders placed at an incidence. The resulting solution is period-doubled and exhibits a triad of spanwise symmetries: a mirror reflection and two spatiotemporal symmetries involving the evolution by half a period (two vortex-shedding cycles) followed by either specular reflection or a half-wavelength shift. The path towards spatio-temporal chaos is initiated thereafter with a modulational period-doubling tertiary bifurcation at $R\in(3.4,3.8)$ that also doubles the spanwise periodicity. The ensuing nonlinear solution repeats only after four vortex shedding periods and retains only a spatiotemporal invariance consisting in the evolution by half a period (two vortex-shedding cycles) followed by mirror reflection about a streamwise-cross-stream plane. At slighlty higher values of $R\geq4$, the flow has become spatio-temporally chaotic, but the main features of mode C are still clearly distinguishable.