Absolute Instability of the Rotating-Disk Boundary Layer

1995 ◽  
pp. 389-396 ◽  
Author(s):  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Rotating Disk ◽  
Absolute Instability

Experimental study of rotating-disk boundary-layer flow with surface roughness

2015 ◽  
Vol 786 ◽  
pp. 5-28 ◽  
Author(s):  
Shintaro Imayama ◽  
P. Henrik Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Excellent Agreement ◽  
Boundary Layer Flow ◽  
Rotating Disk ◽  
Absolute Instability ◽  
Turbulent Transition ◽  
Artificial Surface ◽  
Layer Flow ◽  
Laminar Turbulent Transition

Rotating-disk boundary-layer flow is known to be locally absolutely unstable at $R>507$ as shown by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) and, for the clean-disk condition, experimental observations show that the onset of transition is highly reproducible at that Reynolds number. However, experiments also show convectively unstable stationary vortices due to cross-flow instability triggered by unavoidable surface roughness of the disk. We show that if the surface is sufficiently rough, laminar–turbulent transition can occur via a convectively unstable route ahead of the onset of absolute instability. In the present work we compare the laminar–turbulent transition processes with and without artificial surface roughnesses. The differences are clearly captured in the spectra of velocity time series. With the artificial surface roughness elements, the stationary-disturbance component is dominant in the spectra, whereas both stationary and travelling components are represented in spectra for the clean-disk condition. The wall-normal profile of the disturbance velocity for the travelling mode observed for a clean disk is in excellent agreement with the critical absolute instability eigenfunction from local theory; the wall-normal stationary-disturbance profile, by contrast, is distinct and the experimentally measured profile matches the stationary convective instability eigenfunction. The results from the clean-disk condition are compared with theoretical studies of global behaviours in spatially developing flow and found to be in good qualitative agreement. The details of stationary disturbances are also discussed and it is shown that the radial growth rate is in excellent agreement with linear stability theory. Finally, large stationary structures in the breakdown region are described.

An experimental study of edge effects on rotating-disk transition

2013 ◽  
Vol 716 ◽  
pp. 638-657 ◽  
Author(s):  
Shintaro Imayama ◽  
P. Henrik Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layer Flow ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Global Instability ◽  
Global Mode ◽  
Turbulent Transition ◽  
Layer Flow

AbstractThe 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 rotating-disk 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.

scholarly journals Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices

2017 ◽  
Vol 836 ◽  
pp. 43-71 ◽  
Author(s):  
E. Appelquist ◽  
P. Schlatter ◽  
P. H. Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Low Frequency ◽  
Rotating Disk ◽  
High Amplitude ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Absolute Instability ◽  
Global Instability ◽  
Roughness Elements

This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.

scholarly journals The effects of mass transfer on the global stability of the rotating-disk boundary layer

2010 ◽  
Vol 663 ◽  
pp. 401-433 ◽  
Author(s):  
CHRISTIAN THOMAS ◽  
CHRISTOPHER DAVIES
Keyword(s):  
Boundary Layer ◽  
Mass Transfer ◽  
Rotating Disk ◽  
Disk Surface ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Absolute Instability ◽  
Unstable Region ◽  
Global Behaviour ◽  
Simulation Results

Numerical simulations were conducted to investigate the effects of surface suction and injection on the global behaviour of linear disturbances in the rotating-disk boundary layer. This extends earlier work, which considered the case with no mass transfer. For disturbances in the genuine base flow, where radially inhomogeneity is retained, mass injection at the disk surface led to behaviour that remained qualitatively similar to that which was found when there was no mass transfer. The initial development of disturbances within the absolutely unstable region involved temporal growth and upstream propagation, as should be anticipated for an absolute instability. However, this did not persist indefinitely. Just as for the case without mass transfer, the simulation results suggested that convective behaviour would eventually dominate, for all the Reynolds numbers investigated. In marked contrast, the results obtained for flows with mass suction indicate a destabilization due to the effects of the base-flow radial inhomogeneity. It was possible to identify disturbances excited within the absolutely unstable region that grew continually, with a temporal growth rate that increased as the disturbance evolved. The strong locally stabilizing effect of suction on the absolute instability, which gives rise to large increases in critical Reynolds numbers, appears to be obtainable only at the expense of introducing a new form of global instability. Analogous forms of global behaviour can be found in impulse solutions of the linearized complex Ginzburg–Landau equation. These solutions were deployed to interpret and make comparisons with the numerical simulation results. They illustrate how the long-term behaviour of a disturbance can be determined by the precise balance between radial increases in temporal growth rates, corresponding shifts in temporal frequencies and diffusion/dispersion effects. This balance provides some insight into why disturbances that are absolutely unstable, for the homogenized version of the rotating-disk boundary-layer flow, may become, in the genuine radially inhomogeneous flow, either globally stable or globally unstable, depending on the level of mass transfer that is applied at the disk surface.

Model for unstable global modes in the rotating-disk boundary layer

2010 ◽  
Vol 663 ◽  
pp. 148-159 ◽  
Author(s):  
J. J. HEALEY
Keyword(s):  
Boundary Layer ◽  
Landau Equation ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Global Mode ◽  
Ginzburg Landau ◽  
Varying Coefficients ◽  
Unstable Medium ◽  
Spatially Varying

Recent simulations and experiments appear to show that the rotating-disk boundary layer is linearly globally stable despite the existence of local absolute instability. However, we argue that linear global instability can be created by local absolute instability at the edge of the disk. This argument is based on investigations of the linearized complex Ginzburg–Landau equation with weakly spatially varying coefficients to model the propagation of a wavepacket through a weakly inhomogeneous unstable medium. Therefore, this mechanism could operate in a variety of locally absolutely unstable flows. The corresponding nonlinear global mode has a front at the radius of onset of absolute instability when the disk edge is far from the front. This front moves radially outwards when the Reynolds number at the disk edge is reduced. It is shown that the laminar–turbulent transition front also behaves in this manner, possibly explaining the scatter in observed transitional Reynolds numbers.

scholarly journals The stability and transition of the boundary layer on a rotating sphere

2002 ◽  
Vol 456 ◽  
pp. 199-218 ◽  
Author(s):  
S. J. GARRETT ◽  
N. PEAKE
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Convective Instability ◽  
Reverse Flow ◽  
Experimental Studies ◽  
Rotating Disk ◽  
Absolute Instability ◽  
Mean Flow ◽  
Streamline Curvature ◽  
Curvature Effects

This paper is concerned with convective and absolute instabilities in the boundary-layer flow over the outer surface of a sphere rotating in an otherwise still fluid. Viscous and streamline-curvature effects are included and the analysis is conducted between latitudes of 10° and 80° from the axis of rotation. Both convective and absolute instabilities are found at each latitude within specific parameter spaces. The results of the convective instability analysis show that a crossflow instability mode is the most dangerous below θ = 66°. Above this latitude a streamline-curvature mode is found to be the most dangerous, which coincides with the appearance of reverse flow in the radial component of the mean flow. At low latitudes the disturbances are considered to be stationary, but at higher latitudes they are taken to rotate at 76% of the sphere surface speed, as observed in experimental studies. Our predictions of the Reynolds number and vortex angle at the onset of convective instability are consistent with existing experimental measurements. Results are also presented that suggest that the occurrence of the slowly rotating vortices is associated with the dominance of the streamline-curvature mode at θ = 66°. The local Reynolds number at the predicted onset of absolute instability matches experimental data well for the onset of turbulence at θ = 30°; beyond this latitude the discrepancy increases but remains relatively small below θ = 70°. It is suggested that this absolute instability may cause the onset of transition below θ = 70°. Close to the pole the predictions of each stability analysis are seen to approach those of existing rotating disk investigations.

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