# Large-scale structures in high Reynolds number rotating Waleffe flow

## TL;DR

This study uses direct numerical simulations to analyze large-scale structures and momentum transport mechanisms in high Reynolds number rotating Waleffe flow, revealing optimal rotation conditions and the organization of flow structures.

## Contribution

It provides new insights into the formation and behavior of large-scale structures in rotating Waleffe flow, including the identification of optimal rotation for momentum transport and the comparison with other rotating flows.

## Key findings

- Optimal anti-cyclonic rotation at R_Ω=0.63 minimizes mean energy for given momentum transport.
- Large-scale structures are highly organized in anti-cyclonic regime, similar to rotating plane Couette flow.
- Streamwise vorticity is localized inside the cores of the large-scale rolls.

## Abstract

We perform direct numerical simulations of rotating turbulent Waleffe flow, the flow between two parallel plates with a sinusoidal streamwise shear driving force, to study the formation of large-scale structures and the mechanisms for momentum transport. We simulate different cyclonic and anti-cyclonic rotations in the range of dimensionless rotation numbers (inverse Rossby numbers) $R_\Omega$ $\in$ $[-0.16, 2.21]$, and fix the Reynolds number to $Re=3.16\times 10^3$, large enough such that the shear transport is almost entirely due to Reynolds stresses and viscous transport is negligible. We find an optimum rotation in anti-cyclonic regime at $R_\Omega=0.63$, where a given streamwise momentum transport in the wall-normal direction is achieved with minimum mean energy of the streamwise flow. We link this optimal transport to the strength of large scale structures, as was done in plane Couette by Brauckmann \& Eckhardt (J. Fluid Mech., 815, 2017). Furthermore, we explore the large-scale structures and their behaviour under spanwise rotation, and find disorganized large structures at $R_\Omega =0$ but highly organized structures in the anti-cyclonic regime, similar to the rolls in rotating plane Couette and turbulent Taylor Couette flow. We compare the large scale structures of plane Couette flow and Waleffe flow, and observe that the streamwise vorticity is localized inside the cores of the rolls. We show that the rolls take energy from the mean flow at long time-scales, and relate these structures to eigenvalues of the streamfunction.