Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T13:57:35.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Travelling waves in elliptic pipe flow

Published online by Cambridge University Press:  22 July 2021

Ozge Ozcakir Open the ORCID record for Ozge Ozcakir [Opens in a new window]  and
Philip Hall Open the ORCID record for Philip Hall [Opens in a new window]
Ozge Ozcakir*
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia
Philip Hall
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Families of exact coherent states in elliptical pipe flow obtained from the travelling-wave solutions in circular pipe flow by a continuation approach are found. Results are given in a regime of the aspect ratio $A$ in which the laminar flow is linearly stable. The results suggest the possibility of two distinct classes of solutions of elliptical travelling waves at higher values of $A$: (i) rotationally symmetric centre-mode states that collapse towards the pipe centre and (ii) rotationally asymmetric vortex–wave interaction states with additional mirror symmetry exhibiting organization of the waves around a critical layer. These are the first calculations of three-dimensional travelling waves in elliptical pipes. Investigation of these states has the potential to provide fresh insight into the relationship between exact coherent structures in Poiseuille flow in pipes and channels.

JFM classification

Instability: Nonlinear instability Instability: Transition to turbulence
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , R3
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Davey, A. & Salwen, H. 1994 On the stability of flow in an elliptic pipe which is nearly circular. J. Fluid Mech. 281, 357–369.10.1017/S0022112094003149CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014 Canonical exact coherent structures embedded in high Reynolds number flows. Phil. Trans. R. Soc. A 372 (2020), 20130352.10.1098/rsta.2013.0352CrossRefGoogle ScholarPubMed
Deguchi, K. & Walton, A.G. 2013 A swirling spiral wave solution in pipe flow. J. Fluid Mech. 737, R2.10.1017/jfm.2013.582CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.10.1103/PhysRevLett.91.224502CrossRefGoogle ScholarPubMed
Gibson, J. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.10.1017/jfm.2014.89CrossRefGoogle Scholar
Gibson, J.F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.10.1017/S0022112009990863CrossRefGoogle Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.10.1017/S0022112010002892CrossRefGoogle Scholar
Hall, P. & Smith, F. 1991 On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech. 227, 641–666.10.1017/S0022112091000289CrossRefGoogle Scholar
Hocking, L.M. 1977 The stability of flow in an elliptic pipe with large aspect ratio. Q. J. Mech. Appl. Maths 30, 343–353.10.1093/qjmam/30.3.343CrossRefGoogle Scholar
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), 1594–1598.10.1126/science.1100393CrossRefGoogle ScholarPubMed
Kerswell, R.R. & Davey, A. 1996 On the linear instability of elliptic pipe flow. J. Fluid Mech. 316, 307–324.10.1017/S0022112096000559CrossRefGoogle Scholar
Nagata, M. & Deguchi, K. 2013 Mirror-symmetric exact coherent states in plane Poiseuille flow. J. Fluid Mech. 735, R4.10.1017/jfm.2013.515CrossRefGoogle Scholar
Okino, S. 2011 Nonlinear traveling wave solutions in a square duct flow. PhD thesis, Kyoto University.10.1007/978-3-642-28968-2_28CrossRefGoogle Scholar
Ozcakir, O., Hall, P. & Tanveer, S. 2019 Nonlinear exact coherent structures in pipe flow and their instabilities. J. Fluid Mech. 868, 341–368.10.1017/jfm.2019.20CrossRefGoogle Scholar
Ozcakir, O., Tanveer, S., Hall, P. & Overman, E.A. 2016 Travelling waves in pipe flow. J. Fluid Mech. 791, 284–328.10.1017/jfm.2015.751CrossRefGoogle Scholar
Park, J. & Graham, M. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430–454.10.1017/jfm.2015.554CrossRefGoogle Scholar
Pringle, C.C.T. & Kerswell, R.R. 2007 Asymmetric, helical and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.10.1103/PhysRevLett.99.074502CrossRefGoogle Scholar
Smith, F.T. 1979 Instability of flow through pipes of general cross-section, part 1. Mathematika 26 (2), 187–210.10.1112/S0025579300009761CrossRefGoogle Scholar
Smith, F.T. & Bodonyi, R.J. 1982 Amplitude-dependent neutral modes in the Hagen–Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463–489.Google Scholar
Viswanath, D. 2009 Critical layer in pipe flow at high Reynolds number. Phi. Trans. R. Soc. A 580, 561–576.10.1098/rsta.2008.0225CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.10.1017/S0022112001004189CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 1517–1534.10.1063/1.1566753CrossRefGoogle Scholar
Wedin, H., Bottaro, A. & Nagata, M. 2009 Three-dimensional traveling waves in a square duct. Phys. Rev. E 79 (6 Pt 2), 065305.10.1103/PhysRevE.79.065305CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.10.1017/S0022112004009346CrossRefGoogle Scholar
Supplementary material: File

Ozcakir and Hall supplementary material

Ozcakir and Hall supplementary material

Download Ozcakir and Hall supplementary material(File)
File 101.5 KB