Linear global and asymptotic stability analysis of the flow past rectangular cylinders moving along a wall

Abstract

The primary instability of the steady two-dimensional flow past rectangular cylinders moving parallel to a solid wall is studied, as a function of the cylinder length-to-thickness aspect ratio ${A{\kern-4pt}R} =L/D$ and the dimensionless distance from the wall $g=G/D$ . For all ${A{\kern-4pt}R}$ , two kinds of primary instability are found: a Hopf bifurcation leading to an unsteady two-dimensional flow for $g \ge 0.5$ , and a regular bifurcation leading to a steady three-dimensional flow for $g < 0.5$ . The critical Reynolds number $Re_{c,2\text{-}D}$ of the Hopf bifurcation ( $Re=U_\infty D/\nu$ , where $U_\infty$ is the free stream velocity, $D$ the cylinder thickness and $\nu$ the kinematic viscosity) changes with the gap height and the aspect ratio. For ${A{\kern-4pt}R} \le 1$ , $Re_{c,2\text{-}D}$ increases monotonically when the gap height is reduced. For ${A{\kern-4pt}R} >1$ , $Re_{c,2\text{-}D}$ decreases when the gap is reduced until $g \approx 1.5$ , and then it increases. The critical Reynolds number $Re_{c,3\text{-}D}$ of the three-dimensional regular bifurcation decreases monotonically for all ${A{\kern-4pt}R}$ , when the gap height is reduced below $g < 0.5$ . For small gaps, $g < 0.5$ , the hyperbolic/elliptic/centrifugal character of the regular instability is investigated by means of a short-wavelength approximation considering pressureless inviscid modes. For elongated cylinders, ${A{\kern-4pt}R} > 3$ , the closed streamline related to the maximum growth rate is located within the top recirculating region of the wake, and includes the flow region with maximum structural sensitivity; the asymptotic analysis is in very good agreement with the global stability analysis, assessing the inviscid character of the instability. For cylinders with $AR \leq 3$ , however, the local analysis fails to predict the three-dimensional regular bifurcation.