Global linear instability of the rotating-disk flow investigated through simulations

Appelquist, E. and Schlatter, P. and Alfredsson, P.H. and Lingwood, R.J. (2015) Global linear instability of the rotating-disk flow investigated through simulations. Journal of Fluid Mechanics, 765 (1). pp. 612-631. ISSN 0022-1120

Abstract

Numerical simulations of the flow developing on the surface of a rotating disk are presented based on the linearized incompressible Navier–Stokes equations. The boundary-layer flow is perturbed by an impulsive disturbance within a linear global framework, and the effect of downstream turbulence is modelled by a damping region further downstream. In addition to the outward-travelling modes, inward-travelling disturbances excited at the radial end of the simulated linear region, rend , by the modelled turbulence are included within the simulations, potentially allowing absolute instability to develop. During early times the flow shows traditional convective behaviour, with the total energy slowly decaying in time. However, after the disturbances have reached rend , the energy evolution reaches a turning point and, if the location of rend is at a Reynolds number larger than approximately R=594 (radius non-dimensionalized by ν/Ω∗−−−−√ , where ν is the kinematic viscosity and Ω∗ is the rotation rate of the disk), there will be global temporal growth. The global frequency and mode shape are clearly imposed by the conditions at rend . Our results suggest that the linearized Ginzburg–Landau model by Healey (J. Fluid Mech., vol. 663, 2010, pp. 148–159) captures the (linear) physics of the developing rotating-disk flow, showing that there is linear global instability provided the Reynolds number of rend is sufficiently larger than the critical Reynolds number for the onset of absolute instability.