Bifurcation aspect of polygonal coherence over transitional Reynolds numbers in wide-gap spherical Couette flow

This study numerically investigates the bifurcation aspect of the wide-gap spherical Couette flow (SCF), with an emphasis on the competition among polygonal coherence with different wave numbers observed over transitional Reynolds numbers. Focusing on a representative case, the half-radius ratio $\eta =1/2$, we confirm that the axisymmetric state becomes unstable over the first transitional Reynolds number at which the fourfold spiral state bifurcates, using the continuation method based on the Newton–Raphson algorithm. The Galerkin-spectral method was employed to numerically solve the governing equations. It is found that the threefold spiral state bifurcates from the axisymmetric state at a slightly higher Reynolds number than the first transitional Reynolds number. The attraction of the threefold spiral state expands rapidly with an increase in the Reynolds number, which is determined by verifying the distance of the unstable periodiclike state to both spiral states in the state space. This aspect of the state space explains the experimentally bistable realization of different equilibrium states over the first transitional Reynolds number. This study also found that the periodiclike state is composed of the three- and fourfold spiral states, similar to a beat with two different frequencies.

Figure 1

Configuration of the present study.

Figure 2

SCF experiences the first transition on the route to turbulence in two different manners depending on the value of the geometrical parameter, $\eta$. The solid and dashed curves are the neutral curve of the axisymmetric state against an infinitesimal perturbation of the indexed even and odd azimuthal wave numbers, respectively. The numbers in the circles and triangles indicate the wave numbers of the spiral state observed in previous studies.

Figure 3

Views of (a) polar threefold, (b) equatorial threefold, (c) polar fourfold, and (d) equatorial fourfold.

Figure 4

The dimensionless frequencies against $\text{Re}$, calculated from the phase angular velocities of $m=3$-fold and $m=4$-fold spiral states for $\eta =1/2$, decreased with an increase in $\text{Re}$.

Figure 5

Norm of the antimirror component of $\mathrm{\Phi }$ with $m\ne 0$ plotted against $\text{Re}$ for the three- and fourfold spiral states obtained at $\eta =1/2$.

Figure 6

Time variation of the system at $\text{Re}=490$ projected in the reduced space spanned by $X_3$ and $X_4$. The square on the ordinate indicates the fourfold spiral state bifurcated at $\text{Re}={\text{Re}}_4$ from the axisymmetric state on the origin. The triangle on the abscissa indicates the threefold spiral state performed at $\text{Re}={\text{Re}}_3$. The trajectories indicated by the solid and dashed curves approach the four- and threefold spiral states, respectively.

Figure 7

Time variation of the system at $\text{Re}=560$ projected in the reduced space spanned by $X_3$ and $X_4$. Time variation of the states starting at different initial conditions are indicated as curves. The trajectories indicated by the solid and dashed curves approach the four- and threefold spiral states, respectively.

Figure 8

Time variation of the degree of the fourfold spiral state, $X_4\left(t\right)$, obtained through the edge-tracking at $\text{Re}=560$. Solid curves represent the trajectories starting from an intermediate state ${\mathit{X}}_s$ for $s<s_0$ approaching a threefold spiral state. Dotted curves represent trajectories for $s>s_0$ for a fourfold spiral state.

Figure 9

Views of (a) polar and (b) equatorial twofold obtained at Re = 576.