Local and global instability of the rotating-disk boundary layer of a rotor-stator cavity

This study investigates the stability characteristics of rotating-disk boundary layers in rotor–stator cavities under the frameworks of local linear, global linear and global nonlinear analyses. The local linear stability analysis uses the Chebyshev polynomial method, the global linear stability analysis relies on the linearised incompressible Navier–Stokes (N–S) equations and the global nonlinear analysis involves directly solving the complete incompressible N–S equations. In the local linear framework, the velocity profile derived from the laminar self-similar solution on the rotating-disk side of an infinite rotor–stator cavity is mapped to the Bödewadt–Ekman–von Kármán theoretical model to establish a unified analytical framework. For the global stability study, we extend the methodological framework proposed by Appelquist et al. ( J. Fluid Mech., vol 765, 2015, pp. 612–631) for the von Kármán boundary layer, implementing pulsed disturbances and constructing a radial sponge layer to effectively capture the spatiotemporal evolution of perturbation dynamics while mitigating boundary reflection effects. The analysis reveals that the rotating-disk boundary layer exhibits two distinct instability regimes: convective instability emerges at ${\textit{Re}}=r^*/\sqrt {\nu ^*/\varOmega ^*}=204$ (where $r^*$ is the radius, $\nu ^*$ is the kinematic viscosity and $\varOmega ^*$ is the rotation rate of the system) with azimuthal wavenumber $\beta =27$ , while absolute instability emerges at ${\textit{Re}}=409.6$ with azimuthal wavenumber $\beta =85$ . Under pulsed disturbance excitation, an initial convective instability behaviour dominates in regions exceeding the absolute instability threshold. As perturbations propagate into the sponge layer’s influence domain, upstream mode excitation triggers the emergence of a global unstable mode, characterised by a minimum critical Reynolds number ${\textit{Re}}_{\textit{end}}=484.4$ . Further analysis confirms that this global mode is an inherent property of the rotating-disk boundary layer and is independent of the characteristics of the sponge layer. Frequency-domain analysis establishes that the global mode frequency is governed by local stability characteristics at ${\textit{Re}}_{\textit{end}}$ , while its growth rate evolution aligns with absolute instability trends. By further incorporating nonlinear effects, it was observed that the global properties of the global nonlinear mode remain governed by ${\textit{Re}}_{\textit{end}}$ . The global temporal frequency corresponds to ${\textit{Re}}_{\textit{end}}=471.8$ . When ${\textit{Re}}$ approaches 517.2, the spiral waves spontaneously generate ring-like vortices, which subsequently trigger localised turbulence. This investigation provides novel insights into the fundamental mechanisms governing stability transitions in the rotating-disk boundary layer of the rotor–stator cavity.