# Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist

## TL;DR

This paper investigates the coexistence of Tollmien-Schlichting wave instability and bypass transition routes to turbulence in plane Poiseuille flow using direct numerical simulations, revealing how these routes depend on Reynolds number and initial conditions.

## Contribution

It provides a detailed analysis of the conditions and mechanisms for the coexistence of linear and finite-amplitude transition routes in plane Poiseuille flow, including critical Reynolds numbers and flow structures.

## Key findings

- TS instability occurs at Re ≈ 5815, extending down to Re ≈ 4884.
- Bypass transition appears above Re ≈ 459 with finite-amplitude traveling waves.
- Both routes lead to turbulence but on different time scales.

## Abstract

Plane Poiseuille flow, the pressure driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien-Schlichting (TS) waves, and another one, the bypass transition, that is triggered with finite amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance $2H$ apart and in a domain of width $2\pi H$ and length $2\pi H$ the subcritical instability to TS waves sets in at $Re_{c}=5815$ that extends down to $Re_{TS}\approx4884$. The bypass route becomes available above $Re_E=459$ with the appearance of three-dimensional finite-amplitude traveling waves. The bypass transition covers a large set of finite amplitude perturbations. Below $Re_c$, TS appear for a tiny set of initial conditions that grows with increasing Reynolds number. Above $Re_c$ the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.