Linear stability of flow in a 90° bend

Abstract

This paper considers two-dimensional flow in a channel that consists of straight inlet and outlet branches and a circular 90° curved bend. An incompressible viscous fluid flows through the elbow under the action of a constant pressure gradient between the inlet and outlet. Navier–Stokes equations were solved numerically using a high-fidelity spectral/hp element method. In a range of Reynolds numbers, an adaptive selective frequency damping method was used to obtain steady-state flow. It was found that three separation bubbles and vortex shedding can exist in the bend. The modal stability of two- and three-dimensional perturbations was investigated. The critical Reynolds number of two-dimensional disturbances was found by extrapolation from lower Reynolds number results. It is much greater than the three-dimensional one, but the two-dimensional flow could be subcritically unstable with respect to the externally imposed small-amplitude white noise. For three-dimensional perturbations, the dependence of critical Reynolds numbers on the bending radius was obtained. For the case of a moderate bending radius, a neutral curve is provided and eigenfunctions are studied in detail. Three-dimensional instability can be caused by a periodic or monotonically growing mode, and these unstable modes relate to recirculation bubbles that occur after the bend.