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 = L/D and the dimensionless distance from the wall g = G/D. For all A, two kinds of primary instability are found: a Hopf bifurcation leading to an unsteady two-dimensional flow for g ≥ 0.5, and a regular bifurcation leading to a steady three-dimensional flow for g < 0.5. The critical Reynolds number Rec,2-D of the Hopf bifurcation (Re = U∞D/ν, where U∞ is the free stream velocity, D the cylinder thickness and ν the kinematic viscosity) changes with the gap height and the aspect ratio. For A ≤ 1, Rec,2-D increases monotonically when the gap height is reduced. For A > 1, Rec,2-D decreases when the gap is reduced until g ≈ 1.5, and then it increases. The critical Reynolds number Rec,3-D of the three-dimensional regular bifurcation decreases monotonically for all A, 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 > 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 ≤ 3, however, the local analysis fails to predict the three-dimensional regular bifurcation.