Using stratification to mitigate end effects in quasi-Keplerian Taylor–Couette flow

Abstract

Efforts to model accretion disks in the laboratory using Taylor–Couette flow apparatus are plagued with problems due to the substantial impact the end plates have on the flow. We explore the possibility of mitigating the influence of these end plates by imposing stable stratification in their vicinity. Numerical computations and experiments confirm the effectiveness of this strategy for restoring the axially homogeneous quasi-Keplerian solution in the unstratified equatorial part of the flow for sufficiently strong stratification and moderate layer thickness. If the rotation ratio is too large, however (e.g. ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{3/2}$, where ${\it\Omega}_{o}/{\it\Omega}_{i}$ is the angular velocity at the outer/inner boundary and $r_{i}/r_{o}$ is the inner/outer radius), the presence of stratification can make the quasi-Keplerian flow susceptible to the stratorotational instability. Otherwise (e.g. for ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{1/2}$), our control strategy is successful in reinstating a linearly stable quasi-Keplerian flow away from the end plates. Experiments probing the nonlinear stability of this flow show only decay of initial finite-amplitude disturbances at a Reynolds number $Re=O(10^{4})$. This observation is consistent with most recent computational (Ostilla-Mónico, et al.J. Fluid Mech., vol. 748, 2014, R3) and experimental results (Edlund & Ji, Phys. Rev. E, vol. 89, 2014, 021004) at high $Re$, and reinforces the growing consensus that turbulence in cold accretion disks must rely on additional physics beyond that of incompressible hydrodynamics.

The stratorotational instability of Taylor-Couette flows with moderate Reynolds numbers

“…6. The relation (16) concerns the horizontal lines in this plot for slow-rotation but the values for the fast-rotation lines do hardly differ. The drift rates are uniform in the entire instability cones of Fig.…”

“…In summary, the measured drift values are in good agreement with those derived from the linear stability analysis. Nonlinear simulations for models with endplates suggest that the deviations are produced by the two endplates which have been ignored in formulating the relation (16) but this unexpected result is beyond the scope of this paper. (16).…”

“…Nonlinear simulations for models with endplates suggest that the deviations are produced by the two endplates which have been ignored in formulating the relation (16) but this unexpected result is beyond the scope of this paper. (16). The slope of the dark solid line representing the measurements is 0.77; between the two dashed lines 95% of all measurements are located.…”

“…For strong enough differential rotation this shear layers become unstable and nonaxisymmetric modes develop [3,2]. Adding stratification suppresses this instability and reduces the shear layer and the related circulations [16]. Both effects might appear in the experiment and it is difficult to say how strong the stratification must be in order to stabilize the shear-layer instability.…”

Connections between centrifugal, stratorotational, and radiative instabilities in viscous Taylor-Couette flow

“…The purpose of this paper is to shed some light on these seemingly contradictory statements by carrying out a stability analysis of stratified Taylor-Couette flow which bridges the gap between experimentally-relevant Reynolds numbers and large Reynolds numbers where inviscid analysis should hold in some fashion. The motivation for this study comes from an ongoing programme of experimental work [15][16][17] and the desire to be able to interpret the mechanistic origin of the instabilities observed there. The key questions to be addressed are as follows.…”