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

Published online by Cambridge University Press:  11 December 2017

E. Appelquist Open the ORCID record for E. Appelquist [Opens in a new window] ,
P. Schlatter Open the ORCID record for P. Schlatter [Opens in a new window] ,
P. H. Alfredsson  and
R. J. Lingwood
E. Appelquist
Affiliation:
Linné FLOW Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden Swedish e-Science Research Centre (SeRC), SE-100 44 Stockholm, Sweden
P. Schlatter*
Affiliation:
Linné FLOW Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden Swedish e-Science Research Centre (SeRC), SE-100 44 Stockholm, Sweden
P. H. Alfredsson
Affiliation:
Linné FLOW Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
R. J. Lingwood
Affiliation:
Linné FLOW Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden Division of Engineering Science, School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, London E1 4NS, UK
*
†Email address for correspondence: pschlatt@mech.kth.se
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Abstract

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.

JFM classification

Instability: Absolute/convective instability Instability: Nonlinear instability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 43 - 71
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Copyright
© 2017 Cambridge University Press 

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