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Travelling wave states in pipe flow

  • Ozge Ozcakir (a1), Saleh Tanveer (a2), Philip Hall (a1) and Edward A. Overman (a2)


In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$ . We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$ , though this remains to be confirmed numerically.


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Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14 (4), 529551.
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.
Deguchi, K. & Hall, P. 2014a Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752, 602625.
Deguchi, K. & Hall, P. 2014b Canonical exact coherent structures embedded in high Reynolds number flows. Phil. Trans. R. Soc. Lond. A 372, 20130352.
Deguchi, K. & Walton, A. G. 2013 A swirling spiral wave solution in pipe flow. J. Fluid Mech. 737 (R2), 112.
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Fitzerald, R. 2004 New experiments set the scale for the onset for the onset of turbulence in pipe flow. Phys. Today 57 (2), 2124.
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.
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.
Nagata, M. 1990 Three dimensional finite-amplitude solutions in plane Coutte flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Ozcakir, O.2014 Vortex–wave solutions of Navier–Stokes equations in a cylindrical pipe. PhD thesis, The Ohio State University.
Schneider, T. & Eckhardt, B. 2009 Edge states intermediate between laminar and turbulent dynamics in pipe flow. Phil. Trans. R. Soc. Lond. A 367, 577587.
Smith, F. T. & Bodonyi, R. J. 1982 Amplitude-dependent neutral modes in the Hagen–Poiseille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463489.
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. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883900.
Waleffe, F. 1998 Three dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81 (19), 41404143.
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.
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Travelling wave states in pipe flow

  • Ozge Ozcakir (a1), Saleh Tanveer (a2), Philip Hall (a1) and Edward A. Overman (a2)


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