Flow between rotating disks. Part 2. Stability

Abstract

Infinite-disk flows appear to possess multiple solutions at E−1 = 275 (Holodniok, Kubicek & Hlavacek 1977), where E = ν/s2ω is the Ekman number. One of these solutions exhibits characteristics of Couette flow and is stable in the circular domain 0 < r/s < 50. The other two solutions, both Poiseuille-type flows, are unstable at all positions. The stable solution shows strong resemblance to experimental profiles obtained between finite disks. Stability of finite-disk flows is investigated in two cases: (i) one disk rotating and the other stationary, and (ii) counter-rotating disks. Photographs indicate presence of two instability types. Theoretical calculations are in fair agreement with experimental evidence on instability of type I.

Stability of Bödewadt flow

“…Nevertheless, there remains the possibility that these Type I1 waves are a later stage in the development of the circular waves seen in the central region of the disk in figure 2 (c) (cf. conclusions of Szeri et al 1983). Figure 2(d) shows the developed stage of the flow over the disk.…”

“…Visualization experiments of Clarkson, Chin & Shacter (1980) have provided further data on the nature of Type I instabilities, even though they could not identify Type I1 instabilities in their experiments. During their investigations of the transition of the flow between rotating disks, Szeri et al (1983) reported a type of instability that occurs in irregular patterns. The structure 6.…”

The influence of swirl and confinement on the stability of counterflowing streams

“…Hydrodynamic instability in this particular flow should come as no surprise, given the non-uniqueness of rotating-disk flows (Parter & Rajagopal 1984;Zandbergen & Dijkstra 1987;Goddard et al 1987;Goldshtik & Javorsky 1989;Zhang & Goddard 1989) and the well-known instabilities of Ekman-von Karman boundary layers (Greenspan 1968). Indeed, the stability of flow near rotating disks has been the subject of numerous theoretical investigations over the years, beginning with the works of Stuart and coworkers (Gregory, Stuart & Walker 1955) and including recent works by Szeri, Giron & Schneider (1983), Bodonyi & Ng (1983), Malik (1986) and Faller (1991), to name but a few. These studies, most of which rely on the type of local stability analysis pioneered by Gregory et al (1959, confirm the magnitudes of critical Reynolds numbers observed in various experiments.…”

Flow between rotating disks. Part 1. Basic flow

“…The other members of the family of multiple solutions are unstable at all positions, i.e. at all values of the ratio r / s (Szeri et al 1983).…”

“…A line source or sink of variable strength is placed in coincidence with the axis of rotation. The types of instabilities that may occur in this flow, and the critical conditions for their occurrence, are discussed in the companion paper (Szeri et al 1983).…”