Experimental lateral wall boundary layer behavior of a differentially rotating split-cylinder flow

The cylindrical wall boundary layer of a closed cylinder split in two halves at the equator is studied experimentally. When these two parts rotate in exact corotation the internal flow is essentially in solid-body rotation at the angular velocity of both halves. When a slight difference between the rotation frequencies is established a secondary flow is created due to the differential rotation between both sides and restricted to the boundary layer. This behavior of the boundary layer is compared with theoretical and numerical results finding the “sandwich” structure of a Stewartson boundary layer. Time-dependent waves are observed near the cylindrical wall. Their behavior for different values of the control parameters are presented. Finally, a global recirculation mode is also found due to a symmetry-breaking induced between sides that appears because of a slight misalignment of the experimental setup, whose characteristics are compatible with the behavior of a precessing cylinder.

Figure 1

Cross section of the split cylinder inside the cell (top view). Parts list: 1. Methacrylate semicylinders; 2. Aluminum bases; 3. Stainless steel shafts; 4. External walls of the cell. The axes correspond to the reference frame used, and $\theta$ is the rotation around $z$. $\theta =0$ corresponds to the plane above the symmetry axis.

Figure 2

Time-averaged dimensionless azimuthal velocity (interpolated field) in a horizontal plane (half-planes with $\theta =0^{\circ }$ and ${180}^{\circ }$) at a refined mesh of $91\times 101$ points (one point each mm) for $\text{Re}=1.4\times {10}^4$ and $\text{Ro}=0.2$. There are 25 linearly spaced contour levels in the range $U_{\theta }\in [-1.2,1.2]$, where a positive sign is used to mean that $U_{\theta }$ goes out of the plane, and a negative sign means that $U_{\theta }$ goes into the plane. The white thick curve corresponds to the contour level $U_{\theta }=0$.

Figure 3

PDFs of the dimensionless azimuthal velocity $u_{\theta }$ along a radius at $z=40\mathrm{mm}$ for $\text{Re}=2\times {10}^4$ and $\text{Ro}=0.1$. The dashed line represents the solid-body rotation $u_{\theta }=(1+\text{Ro})r/R$ with a rotation frequency $\mathrm{\Omega }+\omega$, and the dotted line represents the solid-body rotation $u_{\theta }^{\mathrm{solid}}\left(r\right)$. Inset: Detail of the velocity near the cylindrical wall (zoom of the marked rectangle) where the red points correspond to individual velocity measurements, and the hollow black points represent the means of the PDFs.

Figure 4

Azimuthal velocity at $z=40\mathrm{mm}$ and $r=48\mathrm{mm}$ for $\text{Re}=6\times {10}^3$ and $\text{Ro}=0.4$. (a) Azimuthal velocity folded in time and represented along five periods (dashed lines) using the period $2\pi /(\mathrm{\Omega }+\omega )$. (b) PDF of the experimental dimensionless azimuthal velocity (crosses), Gaussians fitting (solid curve), solid-body rotation at $u_{\theta }^{\mathrm{solid}}\left(r_{48}\right)$ (vertical solid line), and solid-body rotation at $(1+\text{Ro})\left(r_{48}/R\right)$ (vertical dashed line).

Figure 5

Dimensionless azimuthal velocities ${\overline{u}}_{\theta }^i$ vs Re for different Ro. Black points represent the velocities at $r=48\mathrm{mm}$ and $z=-40\mathrm{mm}$, and hollow red squares represent the velocities at $r=48\mathrm{mm}$ and $z=40\mathrm{mm}$. The solid line represents the solid-body rotation at $(1-\text{Ro})\left(r_{48}/R\right)$, and the dashed line represents the solid-body rotation at $(1+\text{Ro})\left(r_{48}/R\right)$.

Figure 6

Dimensionless azimuthal velocities ${\overline{u}}_{\theta }^i$ vs Ro along the axial direction at $r=48\mathrm{mm}$ for $\text{Re}=4\times {10}^3$ (top panels) and $\text{Re}=2\times {10}^4$ (bottom panels). The solid line represents the solid-body rotation at $(1-\text{Ro})\left(r_{48}/R\right)$, and the dashed line represents the solid-body rotation at $(1+\text{Ro})\left(r_{48}/R\right)$.

Figure 7

Dimensionless azimuthal velocities ${\overline{u}}_{\theta }^i$ vs the axial position at $r=48\mathrm{mm}$ for different Ro and for $\text{Re}=4\times {10}^3$ (top panels) and $\text{Re}=2\times {10}^4$ (bottom panels). The solid line represents the solid-body rotation at $(1-\text{Ro})\left(r_{48}/R\right)$, and the dashed line represents the solid-body rotation at $(1+\text{Ro})\left(r_{48}/R\right)$.

Figure 8

Dimensionless axial velocity over Ro along a radius at $z=40\mathrm{mm}$ and $\theta =0$ for $\text{Re}=2\times {10}^4$ and $\text{Ro}=0.1$. Hollow black points are the mean axial experimental velocity, the thick green line corresponds to the Kelvin mode, and the green squares are the fluctuations over this mode. The red dotted line represents the theoretical profile using boundary layer theory [27], and the black dashed line represents the null velocity.

Figure 9

Dimensionless azimuthal vorticity $\eta$ created by $U_z^{\prime }$ along a radius at $z=40\mathrm{mm}$ for $\text{Re}=2\times {10}^4$ and $\text{Ro}=0.1$. The black dashed line represents the null vorticity.

Figure 10

Snapshot of the experimental setup using Kalliroscope for $\text{Re}=2\times {10}^4$ and $\text{Ro}=0.1$ for $r<0.63R$. A median filter has been applied to eliminate the noise. The uneven illumination is removed averaging a series of snapshots that covers the timescales of the dynamics. Each snapshot is then divided by this average image to highlight the dynamical behavior.