Linear and nonlinear dynamics of pulsatile channel flow
Résumé
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 flowrate), 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, thus governed by three non-dimensional control parameters. 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 14), the pulsating component is found to have a stabilizing effect, while it is destabilizing at lower frequencies; strongest destabilization is found for Wo ≃ 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 at prescribed total pulsating flowrates. 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 baseflow 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.
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