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.