Threshold exponents of streamwise transition in pipe flows

Subcritical transition in pipe flow is explored for a wide range of Reynolds numbers within the interval ${\rm Re} \in [2.5\times10^3,1.26\times10^4]$ by means of a spectral method that resolves the transitional dynamics with nearly $3.5 \times 10^4$ degrees of freedom on a medium aspect-ratio domain of length $32 \pi /5$. The aim of this exploration is to provide a characterization of the basin of attraction of the basic regime by measuring the minimal amplitude of an initial global perturbation leading to transition. The analysis is based on a particular theoretical scenario that considers streamwise-independent finite amplitude initial vortical perturbations that trigger global transition via optimal inflectional instabilities of streamwise-dependent modes with selected axial wavenumbers. Disturbances consisting of $1, 2$ and $3$ pairs of vortices are investigated. Long lasting turbulent regimes and relaminarized flows are distinguished by means of time integrations of suitable length between $T_{\rm min}=600$ and $T_{\rm max}=1000$ advective time units. Some transitional runs are specifically analized to exemplify the transition scenario under investigation and its independence of pipe length is verified with a few computations on a longer pipe of length $32 \pi$ ($1.4 \times 10^5$ degrees of freedom). For large values of the Reynolds number, a theoretical scaling law for the threshold amplitude of a perturbation required to trigger transition is provided. Different types of perturbations seem to respond to different scaling laws.