Linear and nonlinear dynamics of pulsatile channel flow

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 timeperiodic 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 flowrate 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 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 ≃ 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 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 threedimensional perturbations, the linear and nonlinear dynamics are systematically explored and characterized.

Nonlinear hydrodynamic instability and turbulence in pulsatile flow

“…Another characteristic of the identified mechanism is that the instability occurs only during part of the pulsation cycle, i.e., the deceleration, whereas acceleration relaminarizes the flow. This particular feature is shared with linear modal and nonmodal mechanisms uncovered recently in pulsatile channel flow (23,25). The above findings hence suggest that pulsatile flows of sufficient amplitude, such as cardiovascular flows in large blood vessels, despite being linearly stable can periodically break down into bursts of turbulence.…”

Physiologic blood flow is turbulent

“…Some works have evaluated the stability of monoharmonic (i.e. sinusoidal) flow [32][33][34][35][36] . However, there are no universal criteria that can generally describe the stability of multiharmonic pulsatile flow.…”

Linear Stability Analysis of Pulsatile Quasi-Two-Dimensional Duct Flows under a Transverse Magnetic Field

“…However, due to these additional numerical costs, the size of the governing threeparameter space and the focus on frequencies related to blood flows, the linear stability picture is far from complete. [5] find that the pulsation frequencies which are (most) destabilizing correspond to those found in blood vessels, with stabilization at much higher and lower frequencies. [11] found that the stabilization or destabilization (depending on frequency) was most intense when the maximum velocity of the pulsatile component is of similar magnitude to that of the steady base flow.…”

“…[11] found that the stabilization or destabilization (depending on frequency) was most intense when the maximum velocity of the pulsatile component is of similar magnitude to that of the steady base flow. However, [5] and [11] do not resolve high Reynolds number conditions. In this work, the additional stabilization due to the imposed magnetic field forces computations to much higher Reynolds numbers (order 10 6 for strong fields); see [6] for the steady problem with a transverse magnetic field.…”

“…The linear stability of pulsatile hydrodynamic channel flows has been assessed in [5,11], among others. [5,11] identified the need to validate Floquet methods of stability analysis for pulsatile flows (with time stepping methods) to resolve otherwise dubious or conflicting stability assessments. However, due to these additional numerical costs, the size of the governing threeparameter space and the focus on frequencies related to blood flows, the linear stability picture is far from complete.…”