Skip to main content Accessibility help
×
Home

Nonlinear exact coherent structures in pipe flow and their instabilities

  • Ozge Ozcakir (a1), Philip Hall (a1) and Saleh Tanveer (a2)

Abstract

In this paper, we present computational results of some two-fold azimuthally symmetric travelling waves and their stability. Calculations over a range of Reynolds numbers ( $Re$ ) reveal connections between a class of solutions computed by Wedin & Kerswell (J. Fluid Mech., vol. 508, 2004, pp. 333–371) (henceforth called the WK solution) and the $Re\rightarrow \infty$ vortex–wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In particular, the continuation of the WK solutions to larger values of $Re$ shows that the WK solution bifurcates from a shift-and-rotate symmetric solution, which we call the WK2 state. The WK2 solution computed for $Re\leqslant 1.19\times 10^{6}$ shows excellent agreement with the theoretical $Re^{-5/6}$ , $Re^{-1}$ and $O(1)$ scalings of the waves, rolls and streaks respectively. Furthermore, these states are found to have only two unstable modes in the large $Re$ regime, with growth rates estimated to be $O(Re^{-0.42})$ and $O(Re^{-0.92})$ , close to the theoretical $O(Re^{-1/2})$ and $O(Re^{-1})$ asymptotic results for edge and sinuous instability modes of vortex–wave interaction states (Deguchi & Hall, J. Fluid Mech., vol. 802, 2016, pp. 634–666) in plane Couette flow. For the nonlinear viscous core states (Ozcakir et al., J. Fluid Mech., vol. 791, 2016, pp. 284–328), characterized by spatial a shrinking of the wave and roll structure towards the pipe centre with increasing $Re$ , we continued the solution to $Re\leqslant 8\times 10^{6}$ and we find only one unstable mode in the large Reynolds number regime, with growth rate scaling as $Re^{-0.46}$ within the class of symmetry-preserving disturbances.

Copyright

Corresponding author

Email address for correspondence: ozge.ozcakir@monash.edu

References

Hide All
Blackburn, H. M., Hall, P. & Sherwin, S. J. 2013 Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows. J. Fluid Mech. 726, R2.
Budanur, N. B. & Hof, B. 2018 Complexity of the laminar-turbulent boundary in pipe flow. Phys. Fluids 3, 054401.
Budanur, N. B., Short, K. Y., Farazmand, M., Willis, A. P. & Cvitanovic, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274301.
Chantry, M., Willis, A. P. & Kerswell, R. R. 2014 Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow. Phys. Rev. Lett. 112, 164501.
Deguchi, K. & Hall, P. 2014a The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99112.
Deguchi, K. & Hall, P. 2014b Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752, 602625.
Deguchi, K. & Hall, P. 2016 On the instabilities of vortex–wave interaction states. J. Fluid Mech. 802, 634666.
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008a Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 1141102.
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008b Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243266.
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.
Hall, P. & Smith, F. 1991 On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech. 227, 641666.
Hof, B., van Doorne, C., Westerweel, J., Nieuwstadt, F., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in the turbulent pipe flow. Science 305 (5690), 15941598.
Kerswell, R. & Tutty, O. 2007 Recurrence of traveling waves in transitional pipe flow. J. Fluid Mech. 584, 69102.
Nagata, M. 1990 Three dimensional finite-amplitude solutions in plane Coutte flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Ozcakir, O., Tanveer, S., Hall, P. & Overman, E. A. 2016 Travelling waves in pipe flow. J. Fluid Mech. 791, 284328.
Pringle, C. C. T., Duguet, Y. & Kerswell, R. R. 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. Lond. A 367 (1888), 457472.
Pringle, C. C. T & Kerswell, R. R. 2007 Asymmetric, helical and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.
Soibelman, I. & Meiron, D. 1991 Finite-amplitude bifurcations in plane Poiseuille flow: two-dimensional Hopf bifurcation. J. Fluid Mech. 229, 389416.
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 6776.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358.
Viswanath, D. 2009 Critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. Lond. A 580, 561576.
Viswanath, D. & Cvitanovic, P. 2009 Stable manifolds and the transition to turbulence in pipe flow. J. Fluid Mech. 627, 215233.
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 95 (3), 319343.
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93102.
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 15171534.
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.
Wedin, H. & Kerswell, R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.
Willis, A. P., Duguet, Y., Omel’chenko, O. & Wolfrum, M. 2017 Surfing the edge: using feedback control to find nonlinear solutions. J. Fluid Mech. 831, 579591.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Nonlinear exact coherent structures in pipe flow and their instabilities

  • Ozge Ozcakir (a1), Philip Hall (a1) and Saleh Tanveer (a2)

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