Global stability of the rotating-disk boundary layer

Abstract

The global stability of the von Kármán boundary layer on the rotating disk is reviewed. For the genuine, radially inhomogeneous base flow, linearized numerical simulations indicate that convectively propagating forms of disturbance are predominant at all radii. The presence of absolute instability does not lead to the formation of any unstable linear global mode, even though the temporal growth rate of the absolute instability increases along the radial direction. Analogous behaviour can be found in the impulse solutions of a model amplitude equation, namely the linearized complex Ginzburg–Landau equation. These solutions show that, depending on the precise balance between spatial variations in the temporal growth rate and the corresponding shifts in the temporal frequency, globally stable behaviour can be obtained even in the presence of a strengthening absolute instability. The radial dependency of the absolute temporal frequency is sufficient to detune the disturbance oscillations at different radial positions, thus overcoming the radially increasing absolute growth, thereby giving rise to a stable global response. The origin of this form of behaviour can be traced to the fact that the cylindrical geometry of the rotating-disk flow dictates a choice of a globally valid time non-dimensionalization that, when properly employed, leads to a significant radial variation in the frequency for the absolute instability.