Skip to main content
SpringerLink
Log in

Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer

  • Original Paper
  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

The three-dimensional boundary layer produced by a disk rotating in otherwise still fluid is analytically investigated and its stability properties are systematically established. Using a local parallel flow approximation, finite-amplitude primary travelling vortices governed by a nonlinear dispersion relation are obtained. A secondary stability analysis yields the secondary linear dispersion relation and the secondary absolute growth rate, which determines the long-term stability of the primary nonlinear vortex-trains. By using these local characteristics, spatially developing global patterns of crossflow vortices are derived by employing asymptotic techniques. This approach accounts for both the self-sustained behaviour, exhibiting a sharp transition from laminar to turbulent flow, and the spatial response to external harmonic forcing, for which onset of nonlinearity and transition both depend on the forcing parameters. Based on these results, an open-loop control method is described in detail. Its aim is not to suppress the primary fluctuations but rather to enhance them and to tune them to externally imposed frequency and modenumber, and thereby to delay onset of secondary absolute instability and transition. It is shown that transition can be delayed by more than 100 boundary-layer units.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • von Kármán T (1921) Über laminare und turbulente Reibung. Z Angew Math Mech 1:232–252

    Google Scholar 

  • Reed HL, Saric WS (1989) Stability of three-dimensional boundary layers. Ann Rev Fluid Mech 21:235–284

    Article  ADS  MathSciNet  Google Scholar 

  • Saric WS, Reed HL, White EB (2003) Stability and transition of three-dimensional boundary layers. Annu Rev Fluid Mech 35:413–440

    Article  ADS  MathSciNet  Google Scholar 

  • Schmid PJ, Henningson DS (2001) Stability and transition in shear flows, Applied Mathematical Sciences. Springer, New York

    Google Scholar 

  • Gregory N, Stuart JT, Walker WS (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 248:155–199

    ADS  MathSciNet  MATH  Google Scholar 

  • Huerre P (2000) Open shear flow instabilities. In: Batchelor GK, Moffatt HK, Worster MG (eds) Perspectives in fluid dynamics. Cambridge University Press, Cambridge, pp 159–229

    Google Scholar 

  • Huerre P, Monkewitz PA (1990) Local and global instabilities in spatially developing flows. Ann Rev Fluid Mech 22:473–537

    Article  ADS  MathSciNet  Google Scholar 

  • Lingwood RJ (1995) Absolute instability of the boundary layer on a rotating disk. J Fluid Mech 299:17–33

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Lingwood RJ (1996) An experimental study of absolute instability of the rotating-disk boundary-layer flow. J Fluid Mech 314:373–405

    Article  ADS  Google Scholar 

  • Chomaz J-M, Huerre P, Redekopp LG (1991) A frequency selection criterion in spatially developing flows. Stud Appl Math 84:119–144

    MATH  MathSciNet  Google Scholar 

  • Monkewitz PA, Huerre P, Chomaz J-M (1993) Global linear stability analysis of weakly non-parallel shear flows. J Fluid Mech 251:1–20

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Davies C, Carpenter PW (2003) Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer. J Fluid Mech 486:287–329

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Davies C, Thomas C, Carpenter PW (2007) Global stability of the rotating-disc boundary layer. J Eng Math, doi: 10.1007/s10665-006-9112-8 (this issue)

  • Garrett SJ (2002) The stability and transition of the boundary layer on rotating bodies. Ph.D. thesis, University of Cambridge, Cambridge.

  • Othman H, Corke T (2006) Experimental investigation of absolute instability of a rotating-disk boundary layer. J Fluid Mech 565: 63–94

    Article  MATH  ADS  Google Scholar 

  • Couairon A, Chomaz J-M (1996) Global instabilities in fully nonlinear systems. Phys Rev Lett 77:4015–4018

    Article  ADS  Google Scholar 

  • Couairon A, Chomaz J-M (1997) Absolute and convective instabilities, front velocities and global modes in nonlinear systems. Physica D 108:236–276

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Couairon A, Chomaz J-M (1999) Fully nonlinear global modes in slowly varying flows. Phys Fluids 11:3688–3703

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pier B, Huerre P (1996) Fully nonlinear global modes in spatially developing media. Physica D 97:206–222

    Article  Google Scholar 

  • Pier B, Huerre P (2001) Nonlinear self-sustained structures and fronts in spatially developing wake flows. J Fluid Mech 435:145–174

    Article  MATH  ADS  Google Scholar 

  • Pier B, Huerre P, Chomaz J-M (2001) Bifurcation to fully nonlinear synchronized structures in slowly varying media. Physica D 148:49–96

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Pier B, Huerre P, Chomaz J-M, Couairon A (1998) Steep nonlinear global modes in spatially developing media. Phys Fluids 10:2433–2435

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pier B (2003) Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J Fluid Mech 487:315–343

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Wilkinson SP, Malik MR (1985) Stability experiments in the flow over a rotating disk. AIAA J. 23:588–595

    ADS  Google Scholar 

  • Jarre S, Le Gal P, Chauve MP (1996) Experimental study of rotating disk flow instability. II. Forced flow. Phys Fluids 8:2985–2994

    Article  ADS  Google Scholar 

  • Kohama Y (1984) Study on boundary layer transition of a rotating disk. Acta Mechanica 50:193–199

    Article  ADS  Google Scholar 

  • Pier B (2002) Fully nonlinear waves and transition in the boundary layer over a rotating disk. In: Castro IP, Hancock PE (eds) Advances in Turbulence IX, Proceedings of the 9th European turbulence conference. Barcelona, pp 13–16

  • Pier B (2003) Open-loop control of absolutely unstable domains. Proc R Soc Lond, A 459:1105–1115

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Pier B (2004) Open-loop control and delay of transition in the rotating-disk boundary layer. In: Andersson HI, Krogstad P-A (eds) Advances in Turbulence X, Proceedings of the 10th European turbulence conference. Barcelona, pp 775–778

  • Briggs RJ (1964) Electron-stream interaction with plasmas. M.I.T. Press, Cambridge Mass

    Google Scholar 

  • Bers A (1983) Space-time evolution of plasma instabilities—absolute and convective. In: Rosenbluth M, Sagdeev R (eds) Handbook of plasma physics. North–Holland, Amsterdam, pp 451–517

    Google Scholar 

  • Lingwood RJ (1997) Absolute instability of the Ekman layer and related rotating flows. J Fluid Mech 331:405–428

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Herbert T (1988) Secondary instability of boundary layers. Ann Rev Fluid Mech 20:487–526

    Article  ADS  Google Scholar 

  • Brevdo L, Bridges TJ (1996) Absolute and convective instabilities of spatially periodic flows. Phil Trans R Soc Lond, A 354:1027–1064

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York

    MATH  Google Scholar 

  • Pier B, Huerre P (2001) Nonlinear synchronization in open flows. J Fluids Struct 15:471–480

    Article  Google Scholar 

  • Dee G, Langer JS (1983) Propagating pattern selection. Phys Rev Lett 50:383–386

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de mécanique des fluides et d’acoustique, École centrale de Lyon—CNRS—Université Claude-Bernard Lyon I—INSA, 36 avenue Guy-de-Collongue, 69134, Écully cedex, France

    Benoît Pier

Authors
  1. Benoît Pier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benoît Pier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pier, B. Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J Eng Math 57, 237–251 (2007). https://doi.org/10.1007/s10665-006-9095-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-006-9095-5

Keywords

Search

Navigation