Skip to main content Accessibility help
×
Home

On the impulse response and global instability development of the infinite rotating-disc boundary layer

  • Christian Thomas (a1) and Christopher Davies (a2)

Abstract

Linear disturbance development in the von Kármán boundary layer on an infinite rotating-disc is investigated for an extensive range of azimuthal mode numbers $n$ . The study expands upon earlier investigations that were limited to those values of $n$ located near the onset of absolute instability (Lingwood, J. Fluid Mech., vol. 299, 1995, pp. 17–33), where disturbances to the genuine inhomogeneous flow were shown to be globally stable (Davies & Carpenter, J. Fluid Mech., vol. 486, 2003, pp. 287–329). Numerical simulations corresponding to azimuthal mode numbers greater than the conditions for critical absolute instability display a form of global linear instability that is characterised by a faster than exponential temporal growth, similar in appearance to that found on the rotating-disc with mass suction (Thomas & Davies, J. Fluid Mech., vol. 724, 2010, pp. 510–526) and other globally unstable flows (Huerre & Monkewitz, Annu. Rev. Fluid Mech., vol. 22, 1990, pp. 473–537). Solutions indicate that a change in the global behaviour arises for  $n\in [80:100]$ that is marginally greater than those disturbances studied previously. Furthermore, the Reynolds number associated with the larger azimuthal mode numbers coincides with the upper bound of experimental predictions for transition. Thus, the local–global linear stability of the infinite rotating-disc is similar to the scenario outlined by Huerre & Monkewitz (1990) that states a region of local absolute instability is necessary but not sufficient for global instability to ensue. Conditions are derived to predict the azimuthal mode number needed to bring about a change in global behaviour, based on solutions of the linearised complex Ginzburg–Landau equation coupled with numerical simulations of disturbances to the radially homogeneous flow. The long term response is governed by a detuning effect, based on radial variations of the temporal frequency and matching shifts in temporal growth that increases for larger $n$ , eventually attaining values sufficient to engineer global linear instability. The analysis is extended to include mass transfer through the disc surface, with similar conclusions drawn for disturbances to large enough azimuthal mode numbers. Finally, we conclude that the high $n$ modes are unlikely to have a strong influence on disturbance development and transition in the von Kármán flow, as they will be unable to establish themselves across an extended radial range before nonlinear effects are triggered by the huge growth associated with the wavepacket maxima of the lower $n$ -valued convective instabilities.

Copyright

Corresponding author

Email address for correspondence: christian.thomas@monash.edu

References

Hide All
Appelquist, E., Schlatter, P., Alfredsson, P. H. & Lingwood, R. J. 2015a Global linear instability of the rotating-disk flow investigated through simulations. J. Fluid Mech. 765, 612631.
Appelquist, E., Schlatter, P., Alfredsson, P. H. & Lingwood, R. J. 2015b Investigation of the global instability of the rotating-disk boundary layer. Proc. IUTAM 14, 321328.
Appelquist, E., Schlatter, P., Alfredsson, P. H. & Lingwood, R. J. 2016 On the global nonlinear instability of the rotating-disk flow over a finite domain. J. Fluid Mech. 803, 332355.
Appelquist, E., Schlatter, P., Alfredsson, P. H. & Lingwood, R. J. 2018 Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices. J. Fluid Mech. 836, 4371.
Briggs, R. J. 1964 Electron-Stream Interactions in Plasmas. MIT Press.
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially-developing flows. Phys. Rev. Lett. 60, 2528.
Davies, C. & Carpenter, P. W. 2001 A novel velocity-vorticity formulation of the Navier–Stokes equations with applications to boundary layer disturbance evolution. J. Comput. Phys. 172, 119165.
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. 2017 Global stability behaviour for the BEK family of rotating boundary layers. Theor. Comput. Fluid Dyn. 31, 519536.
Davies, C., Thomas, C. & Carpenter, P. W. 2007 Global stability of the rotating disc boundary layer. J. Engng Maths 57 (3), 219236.
Faller, A. & Kaylor, R. 1966 A numerical study of the instability of the laminar Ekman boundary-layer. J. Atmos. Sci. 23, 466480.
Garrett, S. J. & Peake, N. 2002 The stability and transition of the boundary layer on a rotating sphere. J. Fluid Mech. 456, 405428.
Garrett, S. J. & Peake, N. 2007 The absolute instability of the boundary layer on a rotating cone. Eur. J. Mech. (B/Fluids) 26, 344353.
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.
Gregory, N. & Walker, W. S. 1960 Experiments on the effect of suction on the flow due to a rotating disk. J. Fluid Mech. 9, 225234.
Hannemann, K. & Oertel, H. 1990 Numerical simulation of the absolutely and convectively unstable wake. J. Fluid Mech. 199, 5588.
Harris, D., Bassom, A. P. & Soward, A. M. 2000 An inhomogeneous Landau equation with application to spherical Couette flow in the narrow gap limit. Physica D 137, 260276.
Healey, J. J. 2010 Model for unstable global modes in the rotating-disk boundary layer. J. Fluid Mech. 663, 148159.
Huerre, P. 2000 Open shear flow instabilities. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.). Cambridge University Press.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Hunt, R. E. & Crighton, D. G. 1991 Instability of flows in spatially developing media. Proc. R. Soc. Lond. A 435, 109128.
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2012 A new way to describe the transition characteristics of a rotating-disk boundary-layer flow. Phys. Fluids 24, 031701.
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 On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability. J. Fluid Mech. 745, 132163.
Kobayashi, R., Kohama, Y. & Takamadate, C. 1980 Spiral vortices in boundary layer transition regime on a rotating disk. Acta Mech. 35, 7182.
von Kármán, T. 1921 Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1, 233252.
Leu, T.-Z. & Ho, C.-M. 2000 Control of global instability in a non-parallel near wake. J. Fluid Mech. 404, 345378.
Lingwood, R. & Alfredsson, P. 2015 Instabilities of the von Kármán boundary layer. Appl. Mech. Rev. 67 (3), 030803.
Lingwood, R. J. 1995 Absolute instability of the boundary-layer on a rotating-disk. J. Fluid Mech. 299, 1733.
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. 1997a On the effects of suction and injection on the absolute instability of the rotating-disk boundary layers. Phys. Fluids 9, 13171328.
Lingwood, R. J. 1997b Absolute instability of the Ekman layer and related rotating flows. J. Fluid Mech. 331, 405428.
Mack, L. M.1985 The wave pattern produced by point source on a rotating disk. AIAA Paper 85-0490.
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech. 164, 275287.
Malik, M. R., Wilkinson, S. & Orszag, S. A. 1981 Instability and transition in rotating disk flow. AIAA J. 19, 11311138.
Maxworthy, T. 1999 The flickering candle: transition to a global oscillation in a thermal plume. J. Fluid Mech. 390, 297323 (and Corrigendum 399, 377).
Oertel, H. 1990 Wakes behind blunt bodies. Annu. Rev. Fluid Mech. 22, 539564.
Othman, H. & Corke, T. C. 2006 Experimental investigation of absolute instability of a rotating-disk boundary-layer. J. Fluid Mech. 565, 6394.
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.
Schmid, P. J. & Henningson, D. 2001 Open Shear Flow Instabilities. Springer.
Smith, A. M. O. & Gamberoni, A. H.1956 Transition, pressure gradient and stability theory. Douglas Aircraft Co. Rep. No. ES26388.
Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating sphere. Geophys. Astrophys. Fluid Dyn. 9, 1974.
Soward, A. M. 1992 Thin disc kinematics 𝛼𝜔-dynamo models. Part II. Short length scale modes. Geophys. Astrophys. Fluid Dyn. 64, 201225.
Thomas, C.2007 Numerical simulations of disturbance development in rotating boundary-layers. PhD thesis, Cardiff University.
Thomas, C. & Davies, C. 2010 The effects of mass transfer on the global stability of the rotating-disk boundary laer. J. Fluid Mech. 724, 510526.
Thomas, C. & Davies, C. 2013 Global stability of the rotating-disc boundary layer with an axial magnetic field. J. Fluid Mech. 724, 510526.
Van Ingen, J. L.1956 A suggested semi-empirical method for the calculation of the boundary layer transition region. Dept. Aeronaut. Eng., Delft Univ. Tech., Rep. No. VTH-74.
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.
Zielinska, J. A. & Westfried, J. E. 1995 On the spatial structure of global modes in wake flow. Phys. Fluids 7, 14181424.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

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