Instability driven by boundary inflow across shear: a way to circumvent Rayleigh’s stability criterion in accretion disks?

Abstract

We investigate the two-dimensional (2D) instability recently discussed by Gallet et al. (Phys. Fluids, vol. 22, 2010, 034105) and Ilin & Morgulis (J. Fluid Mech., vol. 730, 2013, pp. 364–378) which arises when a radial cross-flow is imposed on a centrifugally stable swirling flow. By finding a simpler rectilinear example of the instability – a sheared half-plane, the minimal ingredients for the instability are identified and the destabilising/stabilising effect of inflow/outflow boundaries clarified. The instability – christened ‘boundary inflow instability’ here – is of critical layer type where this layer is either at the inflow wall and the growth rate is $O(\sqrt{{\it\eta}})$ (as found by Ilin & Morgulis (J. Fluid Mech., vol. 730, 2013, pp. 364–378)), or in the interior of the flow and the growth rate is $O({\it\eta}\log 1/{\it\eta})$, where ${\it\eta}$ measures the (small) inflow-to-tangential-flow ratio. The instability is robust to changes in the rotation profile, even to those which are very Rayleigh-stable, and the addition of further physics such as viscosity, three-dimensionality and compressibility, but is sensitive to the boundary condition imposed on the tangential velocity field at the inflow boundary. Providing the vorticity is not fixed at the inflow boundary, the instability seems generic and operates by the inflow advecting vorticity present at the boundary across the interior shear. Both the primary bifurcation to 2D states and secondary bifurcations to 3D states are found to be supercritical. Assuming an accretion flow driven by molecular viscosity only, so ${\it\eta}=O(Re^{-1})$, the instability is not immediately relevant for accretion disks since the critical threshold is $O(Re^{-2/3})$ and the inflow boundary conditions are more likely to be stress-free than non-slip. However, the analysis presented here does highlight the potential for mass entering a disk to disrupt the orbiting flow if this mass flux possesses vorticity.