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Optimal energy growth in pulsatile channel and pipe flows

Published online by Cambridge University Press:  06 September 2021

Benoît Pier Open the ORCID record for Benoît Pier [Opens in a new window]  and
Peter J. Schmid Open the ORCID record for Peter J. Schmid [Opens in a new window]
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

Pulsatile channel and pipe flows constitute a fundamental flow configuration with significant bearing on many applications in the engineering and medical sciences. Rotating machinery, hydraulic pumps or cardiovascular systems are dominated by time-periodic flows, and their stability characteristics play an important role in their efficient and proper operation. While previous work has mainly concentrated on the modal, harmonic response to an oscillatory or pulsatile base flow, this study employs a direct–adjoint optimisation technique to assess short-term instabilities, identify transient energy-amplification mechanisms and determine their prevalence within a wide parameter space. At low pulsation amplitudes, the transient dynamics is found to be similar to that resulting from the equivalent steady parabolic flow profile, and the oscillating flow component appears to have only a weak effect. After a critical pulsation amplitude is surpassed, linear transient growth is shown to increase exponentially with the pulsation amplitude and to occur mainly during the slow part of the pulsation cycle. In this latter regime, a detailed analysis of the energy transfer mechanisms demonstrates that the huge linear transient growth factors are the result of an optimal combination of Orr mechanism and intracyclic normal-mode growth during half a pulsation cycle. Two-dimensional sinuous perturbations are favoured in channel flow, while pipe flow is dominated by helical perturbations. An extensive parameter study is presented that tracks these flow features across variations in the pulsation amplitude, Reynolds and Womersley numbers, perturbation wavenumbers and imposed time horizon.

JFM classification

Instability: Shear-flow instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 926 , 10 November 2021 , A11
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Biau, D. 2016 Transient growth of perturbations in Stokes oscillatory flows. J. Fluid Mech. 794, R4.CrossRefGoogle Scholar
Blackburn, H.M., Sherwin, S.J. & Barkley, D. 2008 Convective instability and transient growth in steady and pulsatile stenotic flows. J. Fluid Mech. 607, 267–277.CrossRefGoogle Scholar
Blennerhassett, P.J. & Bassom, A.P. 2002 The linear stability of flat Stokes layers. J. Fluid Mech. 464, 393–410.CrossRefGoogle Scholar
Boyd, J.P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Chiu, J.-J. & Chien, S. 2011 Effects of disturbed flow on vascular endothelium: pathophysiological basis and clinical perspectives. Physiol. Rev. 91, 327–387.CrossRefGoogle ScholarPubMed
Davis, S.H. 1976 The stability of time-periodic flows. Annu. Rev. Fluid Mech. 8, 57–74.CrossRefGoogle Scholar
Gopalakrishnan, S.S., Pier, B. & Biesheuvel, A. 2014 Dynamics of pulsatile flow through model abdominal aortic aneurysms. J. Fluid Mech. 758, 150–179.CrossRefGoogle Scholar
von Kerczek, C.H. 1982 The instability of oscillatory plane Poiseuille flow. J. Fluid Mech. 116, 91–114.CrossRefGoogle Scholar
Ku, D.N. 1997 Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.CrossRefGoogle Scholar
Magri, L. 2019 Adjoint methods as design tools in thermoacoustics. Appl. Mech. Rev. 71, 020801.CrossRefGoogle Scholar
Pedley, T.J. 2000 Blood flow in arteries and veins. In Perspectives in Fluid Dynamics (ed. G.K. Batchelor, H.K. Moffatt & M.G. Worster), pp. 105–158. Cambridge University Press.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. Available at: https://hal.archives-ouvertes.fr/tel-01107517.Google Scholar
Pier, B. & Schmid, P.J. 2017 Linear and nonlinear dynamics of pulsatile channel flow. J. Fluid Mech. 815, 435–480.CrossRefGoogle Scholar
Pier, B. & Schmid, P.J. 2018 Optimal energy growth in pulsatile channel and pipe flows. In 12th European Fluid Mechanics Conference (EFMC12).Google Scholar
Qadri, U., Magri, L., Ihme, M. & Schmid, P.J. 2021 Using adjoint-based optimization to enhance ignition in non-premixed jets. Proc. R. Soc. A 477, 20200472.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Schmid, P.J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle 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
Tuzi, R. & Blondeaux, P. 2008 Intermittent turbulence in a pulsating pipe flow. J. Fluid Mech. 599, 51–79.CrossRefGoogle 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. Physiol. 127, 553–563.CrossRefGoogle ScholarPubMed
Xu, D. & Avila, M. 2018 The effect of pulsation frequency on transition in pulsatile pipe flow. J. Fluid Mech. 857, 937–951.CrossRefGoogle Scholar
Xu, D., Song, B. & Avila, M. 2021 Non-modal transient growth of disturbances in pulsatile and oscillatory pipe flows. J. Fluid Mech. 907, R5.CrossRefGoogle Scholar
Xu, D., Warnecke, S., Song, B., Ma, X. & Hof, B. 2017 Transition to turbulence in pulsating pipe flow. J. Fluid Mech. 831, 418–432.CrossRefGoogle Scholar