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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: [email protected]
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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

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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