On the global nonlinear instability of the rotating-disk flow over a finite domain

Abstract

Direct numerical simulations based on the incompressible nonlinear Navier–Stokes equations of the flow over the surface of a rotating disk have been conducted. An impulsive disturbance was introduced and its development as it travelled radially outwards and ultimately transitioned to turbulence has been analysed. Of particular interest was whether the nonlinear stability is related to the linear stability properties. Specifically three disk-edge conditions were considered; (i) a sponge region forcing the flow back to laminar flow, (ii) a disk edge, where the disk was assumed to be infinitely thin and (iii) a physically realistic disk edge of finite thickness. This work expands on the linear simulations presented by Appelquist et al. (J. Fluid. Mech., vol. 765, 2015, pp. 612–631), where, for case (i), this configuration was shown to be globally linearly unstable when the sponge region effectively models the influence of the turbulence on the flow field. In contrast, case (ii) was mentioned there to be linearly globally stable, and here, where nonlinearity is included, it is shown that both cases (ii) and (iii) are nonlinearly globally unstable. The simulations show that the flow can be globally linearly stable if the linear wavepacket has a positive front velocity. However, in the same flow field, a nonlinear global instability can emerge, which is shown to depend on the outer turbulent region generating a linear inward-travelling mode that sustains a transition front within the domain. The results show that the front position does not approach the critical Reynolds number for the local absolute instability, $R=507$. Instead, the front approaches $R=583$ and both the temporal frequency and spatial growth rate correspond to a global mode originating at this position.

Triboemission of FINE and Ultrafine Aerosol Particles: A New Approach for Measurement and Accurate Quantification

“…The resulting general equation: has an analytical solution in form (27) in the time interval [0; ϕ 0 ] . However, analytical solution of (31) in the time interval [ϕ 0 ; ∞) can be found analytically only if K te,i (t) is a known integrable function. In this case, the general solution of (31) is the following:…”

“…This threshold must rely on the onset of turbulent regime of the air flow. Appelquist et al [31] determined the critical Reynolds number for the onset of global instability of the rotating disk flow Re cr = 583. The transition from transitional to turbulent flow usually occurs for Reynolds numbers between 2300 and 4000.…”

Absolute/convective secondary instabilities and the role of confinement in free shear layers

“…Moreover, in the words of his review [15, p. 385], this "one-step scenario to disorder, involving a global mode made of a wave already absolutely unstable to secondary instability at the global threshold, may also explain the abrupt transition to turbulence observed in the rotating disk..." Chomaz then refers to theoretical and experimental studies by Lingwood, showing that turbulence in rotating disk flow occurs close to where the flow becomes absolutely unstable [37,38], and work by Pier showing that there is an absolute secondary instability in this flow and proposing it as the mechanism of direct transition [39]. Recent investigations [40][41][42] yield support to this scenario, although the secondary instability is not the subharmonic one predicted by Pier [39] and its nature is not yet clear. In any case, the outcome of an absolute secondary instability cannot be predicted a priori; it should be studied on a case by case basis with consideration of global and nonlinear effects.…”

Quantitative Kinetic Analysis of a PdAu3Alloy Catalyst for Oxygen Electro-Reduction

“…Then a discrete set of 40 equidistant values of the disk potential between 0.2 and 0.9 V is chosen. For each disk potential, the disk and ring currents are calculated from simulations and their squared differences to the measured currents at that disk potential are summed up to form the least squares objective function [29] where the bar over theĪ indicates an electrical current normalized to its maximum absolute value over all j (i.e. over all applied E d ).…”