Skip to main content


Log in

Stability and receptivity of boundary layers in a swirl flow channel

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript


The analysis of the disturbances on a spiraling base flow is relevant for the design, operation, and control of technological devices such as parallel-disk turbines and swirl flow channel heat sinks. Spiraling inflow inside an annular cavity closed at the top and bottom is analyzed in the framework of modal and nonmodal stability theories. Local and parallel flow approximations are applied, and the inhomogeneous direction is discretized using the Chebyshev collocation method. The optimal growth of initial disturbances and the optimal response to external harmonic forcing are characterized by the exponential and the resolvent of the dynamics matrix. As opposed to plane Poiseuille flow, transient growth is small, and consequently, it does not play a role in the transition mechanism. The transition is attributed to a crossflow instability that occurs because of the change in the shape of the velocity profile due to rotational effects. Agreement is found between the critical Reynolds number predicted in this work and the deviation of laminar behavior observed in the experiments conducted by Ruiz and Carey (J Heat Transfer 137(7):071702, 2015). For the harmonically driven problem, an energy amplification of \(\textit{O}(100)\) is observed for spiral crossflow waves. Transition to turbulence should be avoided to ensure the safe operation of a parallel-disk turbine, whereas large forcing amplification may be sought to promote mixing in a swirl flow channel heat sink. The analysis presented predicts and provides insight into the transition mechanisms. Due to its easy implementation and low computational cost, it is particularly useful for the early stages of engineering design.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. Do, Y., Lopez, J.M., Marques, F.: Optimal harmonic response in a confined Bödewadt boundary layer flow. Phys. Rev. E 82(3), 036301 (2010)

    Article  Google Scholar 

  2. Gregory, N., Stuart, J.T., Walker, W.S.: On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Philos. Trans. R. Soc. Lond. 248(943), 155–199 (1955)

    Article  MathSciNet  Google Scholar 

  3. Gustavsson, L.H.: Excitation of direct resonances in plane Poiseuille flow. Stud. Appl. Math. 75(3), 227–248 (1986)

    Article  MathSciNet  Google Scholar 

  4. Herrmann-Priesnitz, B., Calderón-Muñoz, W.R., Salas, E.A., Vargas-Uscategui, A., Duarte-Mermoud, M.A., Torres, D.A.: Hydrodynamic structure of the boundary layers in a rotating cylindrical cavity with radial inflow. Phys. Fluids 28(3), 033601 (2016)

    Article  Google Scholar 

  5. Herrmann-Priesnitz, B., Calderón-Muñoz, W.R., Valencia, A., Soto, R.: Thermal design exploration of a swirl flow microchannel heat sink for high heat flux applications based on numerical simulations. Appl. Therm. Eng. 109, 22–34 (2016)

    Article  Google Scholar 

  6. Jovanović, M.R., Bamieh, B.: Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183 (2005)

    Article  MathSciNet  Google Scholar 

  7. Krishnan, V.G., Romanin, V., Carey, V.P., Maharbiz, M.M.: Design and scaling of microscale Tesla turbines. J. Micromech. Microeng. 23(12), 125001 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Lopez, J.M., Marques, F., Rubio, A.M., Avila, M.: Crossflow instability of finite Bödewadt flows: transients and spiral waves. Phys. Fluids 21(11), 114107 (2009)

    Article  Google Scholar 

  10. Malik, M.R.: The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech. 164, 275–287 (1986)

    Article  MathSciNet  Google Scholar 

  11. Orszag, S.A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50(4), 689–703 (1971)

    Article  Google Scholar 

  12. Pfenniger, A., Vogel, R., Koch, V.M., Jonsson, M.: Performance analysis of a miniature turbine generator for intracorporeal energy harvesting. Artif. Organs 38(5), E68 (2014)

    Article  Google Scholar 

  13. Reddy, S.C., Henningson, D.S.: Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238 (1993)

    Article  MathSciNet  Google Scholar 

  14. Ruiz, M., Carey, V.P.: Prediction of single phase heat and momentum transport in a spiraling radial inflow microchannel heat sink. ASME. Paper No. HT2012-58328 (2012)

  15. Ruiz, M., Carey, V.P.: Experimental study of single phase heat transfer and pressure loss in a spiraling radial inflow microchannel heat sink. J. Heat Transfer 137(7), 071702 (2015)

    Article  Google Scholar 

  16. Schmid, P.J.: Nonmodal stability theory. Annu. Rev. Fluid Mech. 39(1), 129–162 (2007)

    Article  MathSciNet  Google Scholar 

  17. Schmid, P.J., Brandt, L.: Analysis of fluid systems: stability, receptivity, sensitivity. Appl. Mech. Rev. 66(2), 024803 (2013)

    Google Scholar 

  18. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, New York (2001)

    Book  Google Scholar 

  19. Sengupta, S., Guha, A.: The fluid dynamics of symmetry and momentum transfer in microchannels within co-rotating discs with discrete multiple inflows. Phys. Fluids 29(9), 093604 (2017)

    Article  Google Scholar 

  20. Serre, E., del Arco, E.C., Bontoux, P.: Annular and spiral patterns in flows between rotating and stationary discs. J. Fluid Mech. 434, 65–100 (2001)

    Article  MathSciNet  Google Scholar 

  21. Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stability without eigenvalues. Science 261(5121), 578–584 (1993)

    Article  MathSciNet  Google Scholar 

Download references


B. H-P. thanks CONICYT-Chile for his Ph.D. scholarship CONICYT-PCHA/Doctorado Nacional/2015-21150139.

Author information

Authors and Affiliations


Corresponding author

Correspondence to B. Herrmann-Priesnitz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Herrmann-Priesnitz, B., Calderón-Muñoz, W.R. & Soto, R. Stability and receptivity of boundary layers in a swirl flow channel. Acta Mech 229, 4005–4015 (2018).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: