The effect of split endcaps on the flow dynamics in a tall Taylor-Couette setup

TL;DR

This work tackles how axially bounded endcaps with a split-ring design modify quasi-Keplerian Taylor-Couette flow and the implications for magnetorotational instability (MRI) experiments. It employs 2D axisymmetric simulations of a tall TC device up to $Re \approx 6\times 10^5$, exploring variations in $\mu$ and analyzing Ekman and Stewartson boundary layers, as well as vortex shedding, complemented by linear MHD stability with an axial field. The study quantifies deviations from the ideal TC profile (up to about $14\%$ in the mid-height angular velocity) and derives boundary-layer scalings: $\delta_{Ek} \sim Re^{-0.52}$ (inner) and $Re^{-0.48}$ (outer); $\delta_{St,w} \sim Re^{-0.25}$ (laminar) transitioning to $Re^{-0.15}$ (turbulent), with $\delta_{St,l} \sim Re^{0.45}$ (laminar) and $Re^{-0.6}$ (turbulent), noting vortex-shedding constraints at high $Re$. The results show endcaps can lower MRI thresholds relative to ideal TC configurations, informing the design and interpretation of DRESDYN-MRI experiments and establishing a foundation for subsequent 3D MHD analyses of endcap-influenced MRI dynamics.

Abstract

The effects of axial boundaries, or endcaps are of fundamental interest in many Taylor-Couette (TC) flow experiments. A main challenge in those experiments has been to minimize these effects, which can substantially alter the flow structure compared to the axially unbounded idealized case. Therefore, understanding and disentangling the influence of endcaps on the TC flow dynamics is essential for the unambiguous interpretation of experimental results, particularly when other dynamical processes (instabilities) in TC flows are involved. In this paper, we study the hydrodynamic evolution of a quasi-Keplerian TC flow in the presence of split endcaps for high Reynolds numbers, $Re$, up to $2\times 10^5$, which are larger than those considered in related previous studies. At these $Re$, the flow deviates from the ideal TC flow profile without endcaps, resulting in about $15\%$ deviation in angular velocity at the mid-height of the cylinders. Aside from turbulent fluctuations caused by shearing instability near the endcaps, the bulk flow remains nearly axially independent and exhibits overall Rayleigh-stability. We characterize the scalings of the Ekman and Stewartson layer sizes with $Re$ as well as examine the effect of the ratio of the outer to inner cylinders' angular velocities on the flow. The implications of these findings for ongoing magnetorotational instability (MRI) experiments based on the similar axially bounded TC setup are also discussed. Specifically, it is shown that when imposing a constant axial magnetic field in all the considered configurations, the flow profile modified by the endcaps lowers the critical threshold for the onset of MRI that in turn can facilitate its emergence and detection in those experiments.

Figures (15)

• Figure 1: (a) TC setup with the split endcaps and (c) its 2D section in the $(r,z)$-plane. The outer wall and its endcap rim are in dark green, while the inner wall and its rim are in red. (b) The radial profile of the azimuthal velocities of the rims (dark green and red) plotted together with that of an unbounded TC flow (light blue).
• Figure 2: Evolution of the volume-averaged non-dimensional kinetic energy at different $Re$. The dashed line denotes the initial energy of the ideal TC flow.
• Figure 3: (a) Deviation, $\varOmega-\varOmega_{\rm TC}$, of the angular velocity $\varOmega$ from the ideal one $\varOmega_{\rm TC}$ in the $(r,z)$-plane. The radial profiles of (b) $\varOmega$ and (c) the specific angular momentum $J_\phi= ru_\phi$ at different $z$ [marked by the horizontal lines in (a)] in the saturated state at $Re=2\times 10^5$.
• Figure 4: Time-averaged radial profiles of (a) the angular velocity $\langle\varOmega\rangle_t$, (b) its relative deviation from the ideal TC profile, $\langle\varOmega/\varOmega_{\rm TC}-1\rangle_t$, and (c) the local shear parameter $\langle q\rangle_t$ (see text) for $Re \in \{10^3, 10^4, 10^5, 2\times 10^5\}$ at the mid-height $z=0$ in the saturated state. The time-averages here and throughout the paper are performed over the time intervals 1620-1717, 3217-3342, 9393-9542 and 11694-11750 (in units of $\varOmega_\mathrm{in}^{-1}$) with the corresponding sampling periods 0.35, 0.45, 0.27 and 0.24 for $Re\in \{10^3, 10^4, 10^5, 2\times 10^5\}$, respectively. For reference, shown are the $q$ profile for the ideal TC flow (dashed) and Rayleigh-stability threshold $q_\mathrm{c}=2$ (dot-dashed). (d) The maximum relative deviation of $\varOmega$ along $r$ at $z=0$ vs. $Re$, which follows $Re^{0.22}$ at high $Re\gtrsim 10^4$.
• Figure 5: Snapshots of the axial $u_z$ (left) and radial $u_r$ (middle) velocities in the $(r,z)$-plane for $Re=10^4$ in the saturated state, exhibiting symmetry between the upper $(z>0)$ and lower ($z<0$) parts of the domain for this $Re$. Ekman circulation direction is shown by the streamlines of the poloidal velocity overplotted on the map of $u_z$. Stable Ekman layers (with thickness $\delta_{Ek}$) at the endcaps are clearly seen in the zoomed-in inset of $u_r$. Right panel shows the corresponding local shear parameter in the $(r,z)$-plane relative to the critical value $q_\mathrm{c}=2$ of Rayleigh-stability, $q-q_\mathrm{c}$. Zoomed-in insets in this panel illustrate the vertical free Stewartson layers originating from the split radius $r_\mathrm{s}$ near the endcaps and characterized by the high shear $q\geq q_\mathrm{c}$ (red). Their width $\delta_{St,w}$ and length $\delta_{St,l}$ are defined, respectively, as the maximum radial and axial extent of the zero level curve of $q-q_\mathrm{c}$ (solid black).