Skip to main content Accessibility help
×
Home

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

  • E. Appelquist (a1) (a2), P. Schlatter (a1) (a2), P. H. Alfredsson (a1) and R. J. Lingwood (a1) (a3)

Abstract

Numerical simulations of the flow developing on the surface of a rotating disk are presented based on the linearized incompressible Navier–Stokes equations. The boundary-layer flow is perturbed by an impulsive disturbance within a linear global framework, and the effect of downstream turbulence is modelled by a damping region further downstream. In addition to the outward-travelling modes, inward-travelling disturbances excited at the radial end of the simulated linear region, $r_{end}$ , by the modelled turbulence are included within the simulations, potentially allowing absolute instability to develop. During early times the flow shows traditional convective behaviour, with the total energy slowly decaying in time. However, after the disturbances have reached $r_{end}$ , the energy evolution reaches a turning point and, if the location of $r_{end}$ is at a Reynolds number larger than approximately $R=594$ (radius non-dimensionalized by $\sqrt{{\it\nu}/{\rm\Omega}^{\ast }}$ , where ${\it\nu}$ is the kinematic viscosity and ${\rm\Omega}^{\ast }$ is the rotation rate of the disk), there will be global temporal growth. The global frequency and mode shape are clearly imposed by the conditions at $r_{end}$ . Our results suggest that the linearized Ginzburg–Landau model by Healey (J. Fluid Mech., vol. 663, 2010, pp. 148–159) captures the (linear) physics of the developing rotating-disk flow, showing that there is linear global instability provided the Reynolds number of $r_{end}$ is sufficiently larger than the critical Reynolds number for the onset of absolute instability.

Copyright

Corresponding author

Email addresses for correspondence: ellinor@mech.kth.se, pschlatt@mech.kth.se

References

Hide All
Appelquist, E.2014 Direct numerical simulations of the rotating-disk boundary-layer flow. Licentiate thesis, Royal Institute of Technology, KTH Mechanics, ISBN: 978-91-7595-202-4.
Bödewadt, U. T. 1940 Die Drehströmung über festem Grund. Z. Angew. Math. Mech. 20, 241253.
Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S.2007 SIMSON – a pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. Royal Institute of Technology, KTH Mechanics, SE-100 44 Stockholm, Sweden.
Davies, C. & Carpenter, P. W. 2003 Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. J. Fluid Mech. 486, 287329.
Davies, C., Thomas, C. & Carpenter, P. W. 2007 Global stability of the rotating-disk boundary layer. J. Engng Maths 57, 219236.
Deville, M. O., Fischer, P. F. & Mund, E. H. 2002 High-Order Methods for Incompressible Fluid Flow. Cambridge University Press.
Ekman, V. W. 1905 On the influence of the Earth’s rotation on ocean currents. Ark. Mat. Astron. Fys. 2 (11), 152.
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G.2012 Nek5000, http://nek5000.mcs.anl.gov.
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155199.
Healey, J. J. 2008 Inviscid axisymmetric absolute instability of swirling jets. J. Fluid Mech. 613, 133.
Healey, J. J. 2010 Model for unstable global modes in the rotating-disk boundary layer. J. Fluid Mech. 663, 148159.
Ho, L.-W.1989 A Legendre spectral element method for simulation of incompressible unsteady viscous free-surface flows. PhD thesis, Massachusetts Institute of Technology.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2013 An experimental study of edge effects on rotating-disk transition. J. Fluid Mech. 716, 638657.
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2014 a On the laminar–turbulent transition of the rotating-disk flow – the role of absolute instability. J. Fluid Mech. 745, 132163.
Imayama, S., Lingwood, R. J. & Alfredsson, P. H. 2014b The turbulent rotating-disk boundary layer. Eur. J. Mech. (B/Fluids) 48, 245253.
von Kármán, T. 1921 Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1, 232252.
Kloker, M. & Konzelmann, U. 1993 Outflow boundary conditions for spatial Navier–Stokes simulations of transition boundary layers. AIAA J. 31, 620628.
Lingwood, R. J. 1995a Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299, 1733.
Lingwood, R. J.1995b Stability and transition of the boundary layer on a rotating disk. PhD thesis, Cambridge University.
Lingwood, R. J. 1996 An experimental study of absolute instability of the rotating-disk boundary-layer flow. J. Fluid Mech. 314, 373405.
Lingwood, R. J. 1997 On the effects of suction and injection on the absolute instability of the rotating-disk boundary layer. Phys. Fluids 9, 13171328.
Littell, H. S. & Eaton, J. K. 1994 Turbulence characteristics of the boundary layer on a rotating disk. J. Fluid Mech. 266, 175207.
Mack, L. M.1985 The wave pattern produced by a point source on a rotating disk. AIAA Paper 85-0490.
Maday, Y. & Patera, A. T. 1989 Spectral element methods for the incompressible Navier–Stokes equations. In State-of-the-Art Surveys on Computational Mechanics (ed. Noor, A. K. & Oden, J. T.), chap. 3, The American Society of Mechanical Engineers.
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech. 164, 275287.
Othman, H. & Corke, T. C. 2006 Experimental investigation of absolute instability of a rotating-disk boundary layer. J. Fluid Mech. 565, 6394.
Patera, A. T. 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468488.
Pier, B. 2003 Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech. 487, 315343.
Pier, B. 2007 Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J. Engng Maths 57, 237251.
Pier, B. 2013 Transition near the edge of a rotating disk. J. Fluid Mech. 737, R1.
Pier, B. & Huerre, P. 2001 Nonlinear synchronization in open flows. J. Fluids Struct. 15, 471480.
Siddiqui, M. E., Mukund, V., Scott, J. & Pier, B. 2013 Experimental characterization of transition region in rotating-disk boundary layer. Phys. Fluids 25, 573576.
Thomas, C. & Davies, C. 2010 The effects of mass transfer on the global stability of the rotating-disk boundary layer. J. Fluid Mech. 663, 401433.
Thomas, C. & Davies, C. 2013 Global stability of the rotating-disc boundary layer with an axial magnetic field. J. Fluid Mech. 724, 510526.
Tufo, H. M. & Fischer, P. F. 2001 Fast parallel direct solvers for coarse grid problems. J. Parallel Distrib. Comput. 61 (2), 151177.
Viaud, B., Serre, E. & Chomaz, J.-M. 2008 The elephant mode between two rotating disks. J. Fluid Mech. 598, 451464.
Viaud, B., Serre, E. & Chomaz, J.-M. 2011 Transition to turbulence through steep global-modes cascade in an open rotating cavity. J. Fluid Mech. 688, 493506.
Wilkinson, S. & Malik, M. R. 1985 Stability experiments in the flow over a rotating disk. AIAA J. 23, 588595.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Related content

Powered by UNSILO

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

  • E. Appelquist (a1) (a2), P. Schlatter (a1) (a2), P. H. Alfredsson (a1) and R. J. Lingwood (a1) (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.