Reynolds number scaling of the influence of boundary layers on the global behavior of laboratory quasi-Keplerian flows

We present fluid velocity measurements in a modified Taylor-Couette device operated in the quasi-Keplerian regime, where it is observed that nearly ideal flows exhibit self-similarity under scaling of the Reynolds number. In contrast, nonideal flows show progressive departure from ideal Couette as the Reynolds number is increased. We present a model that describes the observed departures from ideal Couette rotation as a function of the fluxes of angular momentum across the boundaries, capturing the dependence on Reynolds number and boundary conditions.

Figure 1

Illustration of the HTX device at PPPL showing the inner cylinder in black, the out cylinder in green, and the working fluid in dark blue. The segmented axial boundaries show the inner cylinder extensions in yellow, the independent rings in light blue, and the outer cylinder extensions in green.

Figure 2

Scaled measurements of $v_{\theta }$ at various ${\text{Re}}_s$ for (a) $q=1.8$ and (b) $q=1.5$.

Figure 3

Experiments conducted in $q=1.8$ flows showing $v_{\theta }$ as a function of ${\mathrm{\Omega }}_3$ for the (a) HTX and (b) wide-ring geometries, progressing from low ring speeds (bottom curves) to higher ring speeds (upper curves). In panels (c) and (d) the experimental ${\chi }^2$ (dashed green), for the measured profiles relative to the ideal Coutte profile for these boundary conditions, is plotted against the calculated axial flux of angular momentum from Eq. (2) (red, increasing function) and the pressure differential from Eq. (3) (blue, decreasing function) for the HTX and wide-ring geometries, respectively. ${\mathrm{\Phi }}_z$ is normalized by ${\mathrm{\Phi }}_C$ and $\mathrm{\Delta }p$ is normalized by the kinetic energy density for ideal Couette flow. The vertical dotted lines indicate the zeros of $\mathrm{\Delta }p$ and ${\mathrm{\Phi }}_z$.

Figure 4

Same as described in the cation of Fig. 3 except for $q=1.5$.

Figure 5

Scaling of the velocity differential $\mathrm{\Delta }v$ for the three groups of boundary conditions described in the caption of Fig. 2 for (a) $q=1.8$ and (b) $q=1.5$. The dashed lines in the figure are the solutions to Eqs. (5) and (9), with $\beta =0.08$ and ${\alpha }_{\text{E}}/{\alpha }_{\text{S}}=15$.