An experimental study of edge effects on rotating-disk transition

Abstract

The onset of transition for the rotating-disk flow was identified by Lingwood (J. Fluid. Mech., vol. 299, 1995, pp. 17-33) as being highly reproducible, which motivated her to look for absolute instability of the boundary-layer flow; the flow was found to be locally absolutely unstable above a Reynolds number of 507. Global instability, if associated with laminar-turbulent transition, implies that the onset of transition should be highly repeatable across different experimental facilities. While it has previously been shown that local absolute instability does not necessarily lead to linear global instability: Healey (J. Fluid. Mech., vol. 663, 2010, pp. 148-159) has shown, using the linearized complex Ginzburg-Landau equation, that if the finite nature of the flow domain is accounted for, then local absolute instability can give rise to linear global instability and lead directly to a nonlinear global mode. Healey (J. Fluid. Mech., vol. 663, 2010, pp. 148-159) also showed that there is a weak stabilizing effect as the steep front to the nonlinear global mode approaches the edge of the disk, and suggested that this might explain some reports of slightly higher transition Reynolds numbers, when located close to the edge. Here we look closely at the effects the edge of the disk have on laminar-turbulent transition of the rotatingdisk boundary-layer flow. We present data for three different edge configurations and various edge Reynolds numbers, which show no obvious variation in the transition Reynolds number due to proximity to the edge of the disk. These data, together with the application (as far as possible) of a consistent definition for the onset of transition to others' results, reduce the already relatively small scatter in reported transition Reynolds numbers, suggesting even greater reproducibility than previously thought for 'clean' disk experiments. The present results suggest that the finite nature of the disk, present in all real experiments, may indeed, as Healey (J. Fluid. Mech., vol. 663, 2010, pp. 148-159) suggests, lead to linear global instability as a first step in the onset of transition but we have not been able to verify a correlation between the transition Reynolds number and edge Reynolds number.

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

“…The theoretical R c found by Lingwood (1995a) for the onset of the absolute instability agrees well with the experimentally observed location for the onset of nonlinearity, R = 502-514 (Lingwood 1996) and R = 510-520 (Imayama, Alfredsson & Lingwood 2013), which indicates that the absolute instability is indeed relevant to the transition process. This is particularly interesting since the rotating-disk flow was the first boundary layer found that exhibited absolutely unstable behaviour, as opposed to, say, flat-plate boundary layers, which are convectively unstable.…”

The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream

“…• , respectively), we employ mathematical approximations, specifically using the assumption of large Reynolds number to expand the scale factors, which eventually lead to the expressions (15) and (16). We also focus on the large spiral wavenumber apparent within the problem, which forms the basis for a small parameter expansion in §asymp.…”

The centrifugal instability of the boundary-layer flow over slender rotating cones

“…Although the exact role of local absolute instability in transition over the rotating disk is less clear than originally proposed by Lingwood (1995Lingwood ( ,1996 [see Davies & Carpenter (2003), Pier (2003), Healey (2010), Imayama, Alfredsson & Lingwood (2013)], the theoretical onset of local absolute instability is extremely close to numerous consistent measurements of the onset of turbulence over the rotating disk and this provides a useful of means of comparison. In particular, Garrett & Peake demonstrated that the critical local Reynolds number for local absolute instability is independent of half-angle with R X ≈ 2.5 × 10 5 ; this can be compared to the two sets of experimental results: i) Figure 1 shows the comparison with experimental measurements for the onset of turbulence reported by Kobayashi & Izumi. For cones with ψ ≥ 60…”