  Pulsatile flows are common in nature and in applications, but their stability
and transition to turbulence are still poorly understood. Even in the simple
case of pipe flow subject to harmonic pulsation, there is no consensus among
experimental studies on whether pulsation delays or enhances transition. We
here report direct numerical simulations of pulsatile pipe flow at low
pulsation amplitude A<0.4. We use a spatially localized impulsive disturbance
to generate a single turbulent puff and track its dynamics as it travels
downstream. The computed relaminarization statistics are in quantitative
agreement with the experiments of Xu et al. (J. Fluid Mech., vol. 831, 2017,
pp. 418-432) and support the conclusion that increasing the pulsation amplitude
and lowering the frequency enhance the stability of the flow. In the
high-frequency regime, the behaviour of steady pipe flow is recovered. In
addition, we show that when the pipe length does not permit the observation of
a full cycle, a reduction of the transition threshold is observed. We obtain an
equation quantifying this effect and compare it favourably with the
measurements of Stettler & Hussain (J. Fluid Mech., vol. 170, 1986, pp.
169-197). Our results resolve previous discrepancies, which are due to
different pipe lengths, perturbation methods and criteria chosen to quantify
transition in experiments.
