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Linear and nonlinear dynamics of pulsatile channel flow

Published online by Cambridge University Press:  21 February 2017

Benoît Pier Open the ORCID record for Benoît Pier [Opens in a new window]  and
Peter J. Schmid
Benoît Pier*
Affiliation:
Laboratoire de mécanique des fluides et d’acoustique, CNRS, École centrale de Lyon, Université de Lyon 1, INSA Lyon, 36 avenue Guy-de-Collongue, 69134 Écully, France
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College, South Kensington Campus, London SW7 2AZ, UK
*
†Email address for correspondence: benoit.pier@ec-lyon.fr
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Abstract

The dynamics of small-amplitude perturbations, as well as the regime of fully developed nonlinear propagating waves, is investigated for pulsatile channel flows. The time-periodic base flows are known analytically and completely determined by the Reynolds number $Re$ (based on the mean flow rate), the Womersley number $Wo$ (a dimensionless expression of the frequency) and the flow-rate waveform. This paper considers pulsatile flows with a single oscillating component and hence only three non-dimensional control parameters are present. Linear stability characteristics are obtained both by Floquet analyses and by linearized direct numerical simulations. In particular, the long-term growth or decay rates and the intracyclic modulation amplitudes are systematically computed. At large frequencies (mainly $Wo\geqslant 14$), increasing the amplitude of the oscillating component is found to have a stabilizing effect, while it is destabilizing at lower frequencies; strongest destabilization is found for $Wo\simeq 7$. Whether stable or unstable, perturbations may undergo large-amplitude intracyclic modulations; these intracyclic modulation amplitudes reach huge values at low pulsation frequencies. For linearly unstable configurations, the resulting saturated fully developed finite-amplitude solutions are computed by direct numerical simulations of the complete Navier–Stokes equations. Essentially two types of nonlinear dynamics have been identified: ‘cruising’ regimes for which nonlinearities are sustained throughout the entire pulsation cycle and which may be interpreted as modulated Tollmien–Schlichting waves, and ‘ballistic’ regimes that are propelled into a nonlinear phase before subsiding again to small amplitudes within every pulsation cycle. Cruising regimes are found to prevail for weak base-flow pulsation amplitudes, while ballistic regimes are selected at larger pulsation amplitudes; at larger pulsation frequencies, however, the ballistic regime may be bypassed due to the stabilizing effect of the base-flow pulsating component. By investigating extended regions of a multi-dimensional parameter space and considering both two-dimensional and three-dimensional perturbations, the linear and nonlinear dynamics are systematically explored and characterized.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Instability: Instability Instability: Nonlinear instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 815 , 25 March 2017 , pp. 435 - 480
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© 2017 Cambridge University Press 

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References

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991a An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.Google Scholar
Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991b An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423–444.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat Stokes layers. J. Fluid Mech. 464, 393–410.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2007 The linear stability of high-frequency oscillatory flow in a torsionally oscillating cylinder. J. Fluid Mech. 576, 491–505.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Brevdo, L. & Bridges, T. J. 1997 Absolute and convective instabilities of temporally oscillating flows. Z. Angew. Math. Phys. 48, 290–309.Google Scholar
Conrad, P. W. & Criminale, W. O. 1965 The stability of time-dependent laminar flow: parallel flows. Z. Angew. Math. Phys. 16, 233–254.Google Scholar
Costamagna, P., Vittori, G. & Blondeaux, P. 2003 Coherent structures in oscillatory boundary layers. J. Fluid Mech. 474, 1–33.Google Scholar
Cowley, S. J. 1987 High frequency Rayleigh instability of Stokes layers. In Stability of Time Dependent and Spatially Varying Flows (ed. Dwoyer, D. L. & Hussaini, M. Y.), pp. 261–275. Springer.Google Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Annu. Rev. Fluid Mech. 8, 57–74.Google Scholar
Drazin, P. & Riley, N. 2006 The Navier–Stokes Equations. A Classification of Flows and Exact Solutions. Cambridge University Press.Google Scholar
Ehrenstein, U. & Koch, W. 1991 Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111–148.Google Scholar
Fedele, F., Hitt, D. L. & Prabhu, R. D. 2005 Revisiting the stability of pulsatile pipe flow. Eur. J. Mech. (B/Fluids) 24, 237–254.Google Scholar
Goda, K. 1979 A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows. J. Comput. Phys. 30, 76–95.Google Scholar
Gopalakrishnan, S. S., Pier, B. & Biesheuvel, A. 2014a Dynamics of pulsatile flow through model abdominal aortic aneurysms. J. Fluid Mech. 758, 150–179.Google Scholar
Gopalakrishnan, S. S., Pier, B. & Biesheuvel, A. 2014b Global stability analysis of flow through a fusiform aneurysm: steady flows. J. Fluid Mech. 752, 90–106.Google Scholar
Grosch, C. E. & Salwen, H. 1968 The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177–205.Google Scholar
Hall, P. 1975 The stability of Poiseuille flow modulated at high frequencies. Proc. R. Soc. Lond. A 344, 453–464.Google Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Hall, P. 2003 On the instability of Stokes layers at high Reynolds numbers. J. Fluid Mech. 482, 1–15.Google Scholar
Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193–207.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411–482.Google Scholar
von Kerczek, C. & Davis, S. H. 1974 Linear stability of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.Google Scholar
von Kerczek, C. H. 1982 The instability of oscillatory plane Poiseuille flow. J. Fluid Mech. 116, 91–114.Google Scholar
Ku, D. N. 1997 Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434.Google Scholar
Luo, J. & Wu, X. 2010 On the linear instability of a finite Stokes layer: Instantaneous versus Floquet modes. Phys. Fluids 22, 054106.Google Scholar
McLachlan, M. W. 1964 Theory and Application of Mathieu Functions. Dover.Google Scholar
Merkli, P. & Thomann, H. 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567–575.Google Scholar
Ozdemir, C. E., Hsu, T.-J. & Balachandar, S. 2014 Direct numerical simulations of transition and turbulence in smooth-walled Stokes boundary layer. Phys. Fluids 26, 045108.Google Scholar
Pedley, T. J. 2000 Blood flow in arteries and veins. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 105–158. Cambridge University Press.Google Scholar
Pier, B. 2007 Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J. Engng Maths 57, 237–251.Google Scholar
Pier, B.2015. Dynamique des écoulements ouverts : instabilités et transition, courbure et rotation. Habilitation à diriger des recherches, École centrale de Lyon & Université de Lyon.Google Scholar
Raspo, I., Hugues, S., Serre, E., Randriamampianina, A. & Bontoux, P. 2002 A spectral projection method for the simulation of complex three-dimensional rotating flows. Comput. Fluids 31, 745–767.Google Scholar
Rozhdestvenskii, B. L., Simakin, I. N. & Stoinov, M. I. 1989 Investigation of the stability of a pulsating Poiseuille flow in a planar channel. USSR Comput. Math. Math. Phys. 29, 65–73.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Siggers, J. H. & Waters, S. L. 2008 Unsteady flows in pipes with finite curvature. J. Fluid Mech. 600, 133–165.Google Scholar
Singer, B. A., Ferziger, J. H. & Reed, H. L. 1989 Numerical simulations of transition in oscillatory plane channel flow. J. Fluid Mech. 208, 45–66.CrossRefGoogle Scholar
Stettler, J. C. & Hussain, A. K. M. F. 1986 On transition of the pulsatile pipe flow. J. Fluid Mech. 170, 169–197.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9 (2), 8–106.Google Scholar
Straatman, A. G., Khayat, R. E., Haj-Qasem, E. & Steinman, D. A. 2002 On the hydrodynamic stability of pulsatile flow in a plane channel. Phys. Fluids 14, 1938–1944.Google Scholar
Thomas, C., Bassom, A. P. & Blennerhassett, P. J. 2012 The linear stability of oscillating pipe flow. Phys. Fluids 24, 014106.Google Scholar
Thomas, C., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2010 Direct numerical simulations of small disturbances in the classical Stokes layer. J. Engng Maths 68, 327–338.Google Scholar
Thomas, C., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2011 The linear stability of oscillatory Poiseuille flow in channels and pipes. Proc. R. Soc. Lond. A 467, 2643–2662.Google Scholar
Thomas, C., Blennerhassett, P. J., Bassom, A. P. & Davies, C. 2015 The linear stability of a Stokes layer subjected to high-frequency perturbations. J. Fluid Mech. 764, 193–218.Google Scholar
Thomas, C., Davies, C., Bassom, A. P. & Blennerhassett, P. J. 2014 Evolution of disturbance wavepackets in an oscillatory Stokes layer. J. Fluid Mech. 752, 543–571.CrossRefGoogle Scholar
Tutty, O. R. & Pedley, T. J. 1994 Unsteady flow in a nonuniform channel: A model for wave generation. Phys. Fluids 6, 199–208.Google Scholar
Tuzi, R. & Blondeaux, P. 2008 Intermittent turbulence in a pulsating pipe flow. J. Fluid Mech. 599, 51–79.Google Scholar
Vittori, G. & Verzicco, R. 1998 Direct simulation of transition in an oscillatory boundary layer. J. Fluid Mech. 371, 207–232.Google Scholar
Winter, D. C. & Nerem, R. M. 1984 Turbulence in pulsatile flows. Ann. Biomed. Engng 12, 357–369.Google Scholar
Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when pressure gradient is known. J. Phys. 127, 553–563.Google Scholar
Yang, W. H. & Yih, C.-S. 1977 Stability of time-periodic flows in a circular pipe. J. Fluid Mech. 82, 497–505.Google Scholar