Hysteresis behavior in spanwise rotating plane Couette flow with varying rotation rates

The hysteresis behavior of spanwise rotating plane Couette flow is investigated by direct numerical simulations, where the simulations are carried out at a computational box $8\pi h\times 2h\times 4\pi h$ (here $h$ is the half-channel height) along two opposite directions in rotation number ($\text{Ro}=2\mathrm{\Omega }h/U_w$, with $\mathrm{\Omega }$ the constant angular velocity in the spanwise direction and $U_w$ half of the wall velocity difference) space. When rotation speed increases, structures with two pairs of roll cells appear in $0.02\le \text{Ro}\le 0.3$. However, structures with three pairs of roll cells are observed for $0.03\le \text{Ro}\le 0.3$ when Ro decreases from 0.5. Turbulent statistics on the two branches, such as the friction Reynolds number, turbulent kinetic energy, and kinetic energy from the secondary flows, exhibit similar hysteresis behaviors during $0.03\le \text{Ro}\le 0.3$. Intuitive discussions are also made to explain the hysteresis behavior.

Figure 1

Sketch of the numerical setup in RPCF [39].

Figure 2

Amplitude of the first four spanwise Fourier modes $|\widehat{v}(k_z,t)|$ at the center plane from (a) group 1 and (b) group 2. The data between two vertical dashed lines in (a) and (b) represent the results at a constant rotation speed with the exact value of Ro shown on the top border. (c) and (d) Number of roll-cell pairs $N_z$ represented by the order of the Fourier mode with maximum amplitude in the computation groups, where the horizontal axis in (c) is in logarithmic scale.

Figure 3

Contours of streamwise velocity component $u$ from an instantaneous flow field at $\text{Ro}=0.1$ from group 1: (a) the $x-z$ slice at $y=0$, (b) the $x-z$ slice at $y=-0.8h$, and (c) the $y-z$ slice at $x=L_x/2$.

Figure 4

Contours of streamwise velocity component $u$ from an instantaneous flow field at $\text{Ro}=0.1$ from group 2: (a) the $x-z$ slice at $y=0$, (b) the $x-z$ slice at $y=-0.8h$, and (c) the $y-z$ slice at $x=L_x/2$.

Figure 5

(a) Friction Reynolds number ${\text{Re}}_{\tau }$ at different Ro from different groups. (b) Volume-averaged turbulent kinetic energy $\left[k\right]$ at different Ro from different groups.

Figure 6

Hysteresis behavior of the kinetic energy from the secondary flows.

Figure 7

Growth rate from the linear stability analysis by using the laminar linear velocity as the base flow at (a) $\text{Ro}=0.01$ and (b) $\text{Ro}=0.5$. Here $\alpha$ and $\beta$ are the wave numbers in the streamwise and spanwise directions, respectively.