Transient growth in linearly stable Taylor-Couette flows

TL;DR

This paper investigates the transient energy growth of disturbances in linearly stable Taylor-Couette flows through numerical and analytical methods, revealing universal scaling laws and structures that could influence transition to turbulence.

Contribution

It provides a comprehensive analysis of transient growth across all stable regimes of Taylor-Couette flow, deriving a universal 2/3-scaling law and a semi-empirical estimation formula.

Findings

Energy amplification scales as Re^{2/3} at high Reynolds numbers.

Optimal perturbations become columnar in co-rotating Rayleigh-stable flows.

Transient growth is independent of rotation ratio in the axially invariant limit.

Abstract

Non-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor--Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers Re the optimal transient energy growth always follows a 2/3-scaling with Re, which allows for large amplifications even in regimes where the presence of turbulence remains debated. In co-rotating Rayleigh-stable flows the optimal perturbations become increasingly columnar in their structure, as the optimal axial wavenumber goes to zero. In this limit of axially invariant perturbations we show that linear stability and transient growth are independent of the cylinders' rotation-ratio and we derive a universal 2/3-scaling of optimal energy growth with Re using WKB-theory. Based on this, a semi-empirical formula for the estimation of linear transient growth valid in all regimes is obtained.