Klettner, CA; Smith, FT; (2022) The effect of inertia and vertical confinement on the flow past a circular cylinder in a Hele-Shaw configuration. Journal of Fluid Mechanics , 934 , Article A8. 10.1017/jfm.2021.1128.

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Abstract

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 ( Λ=(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 Λ – ϵ when these conditions are not met. Starting with the Navier–Stokes equations and increasing Λ (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 Λ is increased is the instability of the Poiseuille flow. However, for larger ϵ , this limit is at a much higher Λ , 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 Λ – ϵ .

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