The effect of inertia and vertical confinement on the flow past a circular cylinder in a Hele-Shaw configuration

The Poiseuille flow (centreline velocity $U_c$ ) of a fluid (kinematic viscosity $\nu$ ) past a circular cylinder (radius $R$ ) in a Hele-Shaw cell (height $2h$ ) is traditionally characterised by a Stokes flow ( $\varLambda =(U_cR/\nu )(h/R)^2 \ll 1$ ) through a thin gap ( $\epsilon =h/R \ll 1$ ). In this work we use asymptotic methods and direct numerical simulations to explore the parameter space $\varLambda$ – $\epsilon$ when these conditions are not met. Starting with the Navier–Stokes equations and increasing $\varLambda$ (which corresponds to increasing inertial effects), four successive regimes are identified, namely the linear regime, nonlinear regimes I and II in the boundary layer (the ‘ inner’ region) and a nonlinear regime III in both the inner and outer region. Flow phenomena are studied with extensive comparisons made between reduced calculations, direct numerical simulations and previous analytical work. For $\epsilon =0.01$ , the limiting condition for a steady flow as $\varLambda$ is increased is the instability of the Poiseuille flow. However, for larger $\epsilon$ , this limit is at a much higher $\varLambda$ , resulting in a laminar separation bubble, of size ${O}(h)$ , forming for a certain range of $\epsilon$ at the back of the cylinder, where the azimuthal location was dependent on $\epsilon$ . As $\epsilon$ is increased to approximately 0.5, the secondary flow becomes increasingly confined adjacent to the sidewalls. The results of the analysis and numerical simulations are summarised in a plot of the parameter space $\varLambda$ – $\epsilon$ .

Vortex- and Wake-Induced Vibrations of a Circular Cylinder Placed in the Proximity of Two Fixed Cylinders

Purpose This study aims to understand the vibratory response of a circular cylinder placed in proximity to other fixed bodies. Methods A circular cylinder model was placed in a circulating water channel and was supported elastically to vibrate in the water. Another two circular cylinders were fixed upstream of the vibrating cylinder. The temporal displacement variations of the vibrating cylinder were measured and processed by a frequency analysis. Results When the inline spacings were small, two amplitude peaks appeared in the reduced velocity range 3.0–13.0. When the inline spacings were large, the amplitude response showed a single peak. Conclusion For small inline spacings, the first peak was attributed to high-amplitude vibrations forced by Karman vortex streets shed from the upstream cylinders. The second peak arose from interactions of the wakes of the upstream cylinder with the vibrating cylinder. When the inline spacing increased, the vortex-induced vibrations resembled those of an isolated cylinder.