The development of the flow over a rotating disk is investigated by direct numerical simulations using both the linearized and fully nonlinear incompressible Navier-Stokes equations. These simulations allow investigation of the transition to turbulence of the realistic spatially-developing boundary layer. The current research aims to elucidate further the global linear stability properties of the flow, and relate these to local analysis and discussions in literature. An investigation of the nonlinear upstream (inward) influence is conducted by simulating a small azimuthal section of the disk (1/68). The simulations are initially perturbed by an impulse disturbance where, after the initial transient behaviour, both the linear and nonlinear simulations show a temporally growing upstream mode. This upstream global mode originates in the linear case close to the end of the domain, excited by an absolute instability at this downstream position. In the nonlinear case, it instead originates where the linear region ends and nonlinear harmonics enter the flow field, also where an absolute instability can be found. This upstream global mode can be shown to match a theoretical mode from local linear theory involved in the absolute instability at either the end of the domain (linear case) or where nonlinear harmonics enter the field (nonlinear case). The linear simulation grows continuously in time whereas the nonlinear simulation saturates and the transition to turbulence moves slowly upstream towards smaller radial positions asymptotically approaching a global upstream mode with zero temporal growth rate, which is estimated at a nondimensional radius of 582.
Appelquist, E., Schlatter, P., Alfredsson, P. H., & Lingwood, R. J. (2015). Investigation of the Global Instability of the Rotating-disk Boundary Layer. In Procedia IUTAM (Vol. 14, pp. 321–328). Elsevier. https://doi.org/10.1016/j.piutam.2015.03.054