# TR-2020-07

## Kinetic Energy Flow Instability With Application to Couette Flow

L. Ridgway Scott. 10 August, 2020.

Communicated by L. Ridgway Scott.

### Abstract

We examine a definition of instability for steady flows based on kinetic energy which originated with Reynolds and Orr [37]. Using this definition, we compute the most unstable mode for Couette flow. We find that instability occurs above a Reynolds number of about 177. The usual eigenvalue (linear) stability analysis predicts that there is no instability of Couette flow for any Reynolds number. So the kinetic energy criterion for instability gives a prediction more in accord with experimental evidence. We demonstrate the effect of the perturbation on time-dependent flows. For Reynolds numbers below the critical value, perturbations decay. For Reynolds numbers above the critical value, perturbations grow initially but in all cases studied they eventually decay to zero. However, the maximum amplitude and the persistence time of the perturbations both increase with Reynolds number in all cases studied.

### Original Document

The original document is available in PDF (uploaded 10 August, 2020 by
L. Ridgway Scott).