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Non-modal transient growth of disturbances in pulsatile and oscillatory pipe flows

Published online by Cambridge University Press:  02 December 2020

Duo Xu Open the ORCID record for Duo Xu [Opens in a new window] ,
Baofang Song Open the ORCID record for Baofang Song [Opens in a new window]  and
Marc Avila Open the ORCID record for Marc Avila [Opens in a new window]
Duo Xu*
Affiliation:
Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, 28359Bremen, Germany
Baofang Song
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin300072, PR China
Marc Avila
Affiliation:
Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, 28359Bremen, Germany
*
†Email address for correspondence: duo.xu@zarm.uni-bremen.de
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Abstract

Laminar flows through pipes driven at steady, pulsatile or oscillatory rates undergo a subcritical transition to turbulence. We carry out an extensive linear non-modal stability analysis of these flows and show that for sufficiently high pulsation amplitudes the stream-wise vortices of the classic lift-up mechanism are outperformed by helical disturbances exhibiting an Orr-like mechanism. In oscillatory flow, the energy amplification depends solely on the Reynolds number based on the Stokes-layer thickness, and for sufficiently high oscillation frequency and Reynolds number, axisymmetric disturbances dominate. In the high-frequency limit, these axisymmetric disturbances are exactly similar to those recently identified by Biau (J. Fluid Mech., vol. 794, 2016, R4) for oscillatory flow over a flat plate. In all regimes of pulsatile and oscillatory pipe flow, the optimal helical and axisymmetric disturbances are triggered in the deceleration phase and reach their peaks in typically less than a period. Their maximum energy gain scales exponentially with Reynolds number of the oscillatory flow component. Our numerical computations unveil a plausible mechanism for the turbulence observed experimentally in pulsatile and oscillatory pipe flow.

JFM classification

Instability: Nonlinear instability Instability: Transition to turbulence
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , R5
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 1435–1458.CrossRefGoogle Scholar
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
Boyd, J. P 1983 The continuous spectrum of linear Couette flow with the beta effect. J. Atmos. Sci. 40 (9), 2304–2308.2.0.CO;2>CrossRefGoogle Scholar
Chiu, J.-J. & Chien, S 2011 Effects of disturbed flow on vascular endothelium: pathophysiological basis and clinical perspectives. Physiol. Rev. 91 (1), 327–387.CrossRefGoogle ScholarPubMed
Eckmann, D. M. & Grotberg, J. B. 1991 Experiments on transition to turbulence in oscillatory pipe flow. J. Fluid Mech. 222, 329–350.CrossRefGoogle Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flows. Phys. Fluids 31, 2093–2102.CrossRefGoogle Scholar
Feldmann, D. & Wagner, C 2012 Direct numerical simulation of fully developed turbulent and oscillatory pipe flows at $Re_{\tau }=1440$. J. Turbul. 13 (32), 1–28.CrossRefGoogle Scholar
Feldmann, D. & Wagner, C. 2016 On the influence of computational domain length on turbulence in oscillatory pipe flow. Intl J. Heat Fluid Flow 61, 229–244.CrossRefGoogle Scholar
Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193–207.CrossRefGoogle Scholar
Kleinstreuer, C. & Zhang, Z. 2010 Airflow and particle transport in the human respiratory system. Annu. Rev. Fluid Mech. 42, 301–334.CrossRefGoogle Scholar
Ku, D. N 1997 Blood flow in arteries. Annu. Rev. Fluid Mech. 29 (1), 399–434.CrossRefGoogle Scholar
Maretzke, S., Hof, B. & Avila, M. 2014 Transient growth in linearly stable Taylor–Couette flows. J. Fluid Mech. 742, 254–290.CrossRefGoogle Scholar
Merkli, P. & Thomann, H. 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567–575.CrossRefGoogle Scholar
Meseguer, Á. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number $10^7$. J. Comput. Phys. 186, 178–197.CrossRefGoogle Scholar
Nebauer, J. R. A. 2019 On the stability and transition of time-periodic pipe flow. PhD thesis, Monash University.Google Scholar
Orr, W. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part 2. A viscous liquid. Proc. R. Irish Acad. 27, 69–138.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935–982.Google Scholar
Sarpkaya, T. 1966 Experimental determination of the critical Reynolds number for pulsating Poiseuille flow. Trans. ASME: J. Fluids Engng 88, 589–598.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sergeev, S. I. 1966 Fluid oscillations in pipes at moderate Reynolds numbers. Fluid Dyn. 1, 21–22.Google Scholar
Stettler, J. & Hussain, A. 1986 On transition of the pulsatile pipe flow. J. Fluid Mech. 170, 169–197.CrossRefGoogle Scholar
Thomas, C., Bassom, A. P. & Blennerhassett, P. J. 2012 The linear stability of oscillating pipe flow. Phys. Fluids 24, 014106.CrossRefGoogle Scholar
Thomas, C., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2011 The linear stability of oscillatory Poiseuille flow in channels and pipes. Phil. Trans. R. Soc. Lond. A 467, 2643–2662.Google Scholar
Tsigklifis, K. & Lucey, A. D. 2017 Asymptotic stability and transient growth in pulsatile Poiseuille flow through a compliant channel. J. Fluid Mech. 820, 370–399.CrossRefGoogle Scholar
Von Kerczek, C. H. 1982 The instability of oscillatory plane Poiseuille flow. J. Fluid Mech. 116, 91–114.CrossRefGoogle Scholar
Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient in 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., Varshney, A., Ma, X., Song, B., Riedl, M., Avila, M. & Hof, B 2020 Nonlinear hydrodynamic instability and turbulence in pulsatile flow. Proc. Natl Acad. Sci. USA 117 (21), 11233–11239.CrossRefGoogle ScholarPubMed
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
Zhao, T. S. & Cheng, P 1996 Experimental studies on the onset of turbulence and frictional losses in an oscillatory turbulent pipe flow. Intl J. Heat Fluid Flow 17 (4), 356–362.CrossRefGoogle Scholar

Xu et al. supplementary movie 1

Dynamics of the optimal classic disturbance in pulsatile pipe flow (shown in figure 1d—e); $Re_s=2000$, $A=1$ and $Wo=15$. The left panel shows contours of the stream-wise vorticity on a $r-\theta$ cross-section, the right panel shows the contours of the stream-wise velocity on a $z-r$ cross-section. The disturbance is initialized at $t_0/T=0.25$ and reaches its peaked energy gain at $t_f/T=1.75$.
Download Xu et al. supplementary movie 1(Video)
Video 1.8 MB

Xu et al. supplementary movie 2

Dynamics of the optimal helical disturbance in pulsatile pipe flow (shown in figure 1f—g); $Re_s=2000$, $A=1$ and $Wo=15$. The left panel shows contours of the stream-wise vorticity on a $r-\theta$ cross-section, the right panel shows the contours of the stream-wise velocity on a $z-r$ cross-section. The disturbance is initialized at $t_0/T=0.5$ and peaks in energy at $t_f/T=1.2$.
Download Xu et al. supplementary movie 2(Video)
Video 715 KB

Xu et al. supplementary movie 3

Dynamics of the optimal helical disturbance in oscillatory pipe flow (shown in figure 6d—g); $Re_\delta=530$ and $Wo=10$. The left panel shows the contours of the stream-wise vorticity on a $r-\theta$ cross-section, the right panel shows the contours of the span-wise vorticity on a $z-r$ cross-section. The disturbance is initialized at $t_0/T=0.35$ and peaks in energy at $t_f/T=0.75$.
Download Xu et al. supplementary movie 3(Video)
Video 4.1 MB

Xu et al. supplementary movie 4

Dynamics of the optimal axisymmetric disturbance in oscillatory pipe flow (shown in figure 6h—k); $Re_\delta=589$ and $Wo=15$. The left panel shows the contours of the stream-wise vorticity on a $r-\theta$ cross-section, the right panel shows the contours of the span-wise vorticity on a $z-r$ cross-section. The disturbance is initialized at $t_0/T=0.35$ and peaks in energy at $t_f/T=0.75$.
Download Xu et al. supplementary movie 4(Video)
Video 7.7 MB