Self-sustaining process in Taylor-Couette flow

The transition from Tayor vortex flow to wavy-vortex flow is revisited. The self-sustaining process (SSP) of Waleffe [Phys. Fluids 9, 883 (1997)] proposes that a key ingredient in transition to turbulence in wall-bounded shear flows is a three-step process involving rolls advecting streamwise velocity, leading to streaks which become unstable to a wavy perturbation whose nonlinear interaction with itself feeds the rolls. We investigate this process in Taylor-Couette flow. The instability of Taylor-vortex flow to wavy-vortex flow, a process which is the inspiration for the second phase of the SSP, is shown to be caused by the streaks, with the rolls playing a negligible role, as predicted by Jones [J. Fluid Mech. 157, 135 (1985)] and demonstrated by Martinand et al. [Phys. Fluids 26, 094102 (2014)]. In the third phase of the SSP, the nonlinear interaction of the waves with themselves reinforces the rolls. We show this both quantitatively and qualitatively, identifying physical regions in which this reinforcement is strongest, and also demonstrate that this nonlinear interaction depletes the streaks.

Figure 1

(a) Dependence of critical Reynolds number ${\text{Re}}^{\mathrm{TVF}}$ on the radius ratio $\eta$ for transition to Taylor-vortex flow (right scale). Also shown is the relationship between azimuthal wave number $M_0$ and $\eta$ for circumferential wavelength $L_{\theta }=5$ (left scale). Our chosen parameter values are $\eta =0.92$ and $M_0=15$. (b) Schematic decomposition of flow into ${\mathbf{U}}_{\mathrm{Cou}},{\mathbf{U}}_{\mathrm{mean}},{\mathbf{U}}_{\mathrm{roll}},{\mathbf{U}}_{\mathrm{streak}},{\mathbf{U}}_{\mathrm{wave}}$ according to axial and azimuthal Fourier modes $k$ and $m$.

Figure 2

Visualizations in the meridional $(r,z)$ plane of (a) Taylor-vortex flow (without laminar Couette flow), (b) the $M_0=15$ eigenvector leading to wavy-vortex flow, and (c) nonlinear interaction of this eigenvector with itself. The parameters are $\text{Re}=300$ and $\eta =0.92$. The inner cylinder is on the left, and the outer cylinder on the right. In each case, the meridional velocity within the plane is indicated by arrows, and the azimuthal velocity perpendicular to it is indicated by colors. Red indicates a positive deviation of the azimuthal velocity from laminar Couette flow, blue a negative deviation, and green no deviation. Thus, in panel (a) the arrows show the rolls and the colors show the streaks of Taylor-vortex flow. The white dashed boxes in panels (c) and (a) highlight the alignment between the axial components (arrows) of $\langle {\mathbf{u}}_{\mathrm{wvf}}\times \mathbf{\nabla }\times {\mathbf{u}}_{\mathrm{wvf}}\rangle$ and of the rolls of ${\mathbf{U}}_{\mathrm{TVF}}$, which comprise the third step of the SSP.

Figure 3

Wavy-vortex flow (including Taylor-vortex flow but not laminar Couette flow) at $\text{Re}=300$. Above: four meridional planes over azimuthal period $[0,2\pi /15]$. Azimuthal velocity indicated by colors, meridional velocity by arrows. Below: one azimuthal period $[0,2\pi /15]$ at midgap $\overline{r}=12$. Radial velocity indicated by colors. The dashed lines indicate the positions of the four meridional planes shown above.

Figure 4

Comparison of linearization about ${\mathbf{U}}_{\text{TVF}}$ and about ${\mathbf{U}}_{\text{TVF}}-{\mathbf{U}}_{\text{roll}}$. Eigenvectors ${\mathbf{u}}_{\text{wvf}}$ resulting from linearization about (a) only ${\mathbf{U}}_{\text{TVF}}-{\mathbf{U}}_{\text{roll}}$ and (b) the full ${\mathbf{U}}_{\text{TVF}}$ for $\text{Re}=300$. Azimuthal velocity designated by color (red for positive, blue for negative, green for zero) and radial and axial velocity by arrows. (c) Growth rate (real part of eigenvalue) for linearization about ${\mathbf{U}}_{\text{TVF}}$ (black, solid), ${\mathbf{U}}_{\text{TVF}}-{\mathbf{U}}_{\text{roll}}$ (blue, dashed), as a function of $Re$. Since omitting ${\mathbf{U}}_{\mathrm{roll}}$ from the base flow barely changes the eigenvector or eigenvalue, it is clear that it plays no role in the instability.

Figure 5

(a) Energy decomposition for ${\mathbf{U}}_{\mathrm{TVF}}$ and ${\mathbf{U}}_{\mathrm{WVF}}$. [See Fig. 1 for definitions of this decomposition.] Curves marked with crosses correspond to the components of ${\mathbf{U}}_{\mathrm{TVF}}$ originating at $\text{Re}=146$. Curves marked with circles correspond to the energy components of ${\mathbf{U}}_{\mathrm{WVF}}$, which bifurcates at $\text{Re}=201$. The streak energy is lower for WVF than it is for TVF; the difference between the two is close to the energy in the waves (which is necessarily zero for TVF). The energy in the deviation of the mean from Couette flow is also lower for WVF than for TVF. The energy in the rolls is approximately the same for the two flows. (b) Normalized inner product of nonlinear self-interaction $\langle \mathbf{NL},{\mathbf{U}}_{\mathrm{roll}}\rangle /\left|\right|{\mathbf{U}}_{\mathrm{roll}}\left|\right|,\langle \mathbf{NL},{\mathbf{U}}_{\mathrm{streak}}\rangle /\left|\right|{\mathbf{U}}_{\mathrm{streak}}\left|\right|$ and $\langle \mathbf{NL},{\mathbf{U}}_{\mathrm{mean}}\rangle /\left|\right|{\mathbf{U}}_{\mathrm{mean}}\left|\right|$ for rolls, streaks, and deviation of the mean from Couette flow. The nonlinear term $\mathbf{NL}$ feeds the rolls and mean but drains the streaks.