Hysteresis Behavior of Large Scale Structure in Turbulent Rotating Plane Couette Flow

The existence of counter rotating vortex pairs is important in rotating plane Couette flow (RPCF). Two groups of numerical simulation were conducted with rotation number, $Ro$ ($Ro=2\Omega h/U_w$, with $\Omega$ being the constant angular velocity in the spanwise direction, $h$ being half channel height and $U_w$ being half wall velocity difference), varying between 0.01 and 0.6. We found that the number of vortex pairs in a finite computation domain exhibits a hysteretic behavior as $Ro$ increases and decreases. When $Ro$ increases from 0.03 to 0.3, the number of vortex pairs is 2. When $Ro$ decreases from 0.3 to 0.03, the number of vortex pairs is 3. This phenomenon is related to multiple states in RPCF or state bifurcation. Turbulent statistics such as friction, turbulent kinetic energy also form hysteresis loops. A linear stability analysis is performed on this problem to explain the formation of the hysteresis behavior.