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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: [email protected]
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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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