# Analysis and modeling of localized invariant solutions in pipe flow

## TL;DR

This paper investigates the spatial structure of localized turbulence in pipe flow, modeling decay rates and velocities of turbulent fronts using linearized Navier-Stokes equations, and validates the model against turbulent puffs.

## Contribution

It introduces a linearized modeling approach for the decay and velocity profiles of localized turbulent structures in pipe flow, linking kinematics with spatial structure.

## Key findings

- Exponential decay of turbulent fronts towards laminar flow
- Model accurately predicts decay rates and front velocities
- Validates model with turbulent puff data

## Abstract

Turbulent spots surrounded by laminar flow are a landmark of transitional shear flows, but the dependence of their kinematic properties on spatial structure is poorly understood. We here investigate this dependence in pipe flow for Reynolds numbers between 1500 and 5000. We compute spatially localized relative periodic orbits in long pipes and show that their upstream and downstream fronts decay exponentially towards the laminar profile. This allows to model the fronts by employing the linearized Navier-Stokes equations, and the resulting model yields the spatial decay rate and the front velocity profiles of the periodic orbits as a function of Reynolds number, azimuthal wave number and propagation speed. In addition, when applied to a localized turbulent puff, the model is shown to accurately approximate the spatial decay rate of its upstream and downstream tails. Our study provides insight into the relationship between the kinematics and spatial structure of localized turbulence and more generally into the physics of localization.