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

TL;DR

This paper performs a linear stability analysis of pulsatile and oscillatory pipe flows, revealing new disturbance mechanisms and scaling laws that explain turbulence onset in these flows.

Contribution

It introduces a comprehensive analysis of non-modal disturbances, identifying helical and axisymmetric modes as dominant under different conditions, and links these to turbulence mechanisms.

Findings

Helical disturbances outperform stream-wise vortices at high pulsation amplitudes.

Axisymmetric disturbances dominate at high oscillation frequencies and Reynolds numbers.

Maximum energy gain scales exponentially with Reynolds number.

Abstract

Laminar flows through pipes driven at steady, pulsatile or oscillatory rates undergo a sub-critical 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 (2016) 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.