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

The Poiseuille flow (centreline velocity Uc ) of a fluid (kinematic viscosity ν ) past a circular cylinder (radius R ) in a Hele-Shaw cell (height $2h$ ) is traditionally characterised by a Stokes flow ( =(UcR/ν )(h/R)² 1 ) through a thin gap ( ε =h/R 1 ). In this work we use asymptotic methods and direct numerical simulations to explore the parameter space – ε when these conditions are not met. Starting with the Navier–Stokes equations and increasing (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 ε =0.01 , the limiting condition for a steady flow as is increased is the instability of the Poiseuille flow. However, for larger ε , this limit is at a much higher , resulting in a laminar separation bubble, of size O(h) , forming for a certain range of ε at the back of the cylinder, where the azimuthal location was dependent on ε . As ε 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 – ε .