We present azimuthal velocity profiles measured in a Taylor-Couette apparatus, which has been used as a model of stellar and planetary accretion disks. The apparatus has a cylinder radius ratio of , an aspect-ratio of , and the plates closing the cylinders in the axial direction are attached to the outer cylinder. We investigate angular momentum transport and Ekman pumping in the Rayleigh-stable regime. The regime is linearly stable and is characterized by radially increasing specific angular momentum. We present several Rayleigh-stable profiles for shear Reynolds numbers , both for (quasi-Keplerian regime) and (sub-rotating regime) where is the inner/outer cylinder rotation rate. None of the velocity profiles matches the non-vortical laminar Taylor-Couette profile. The deviation from that profile increased as solid-body rotation is approached at fixed . Flow super-rotation, an angular velocity greater than that of both cylinders, is observed in the sub-rotating regime. The velocity profiles give lower bounds for the torques required to rotate the inner cylinder that were larger than the torques for the case of laminar Taylor-Couette flow. The quasi-Keplerian profiles are composed of a well mixed inner region, having approximately constant angular momentum, connected to an outer region in solid-body rotation with the outer cylinder and attached axial boundaries. These regions suggest that the angular momentum is transported axially to the axial boundaries. Therefore, Taylor-Couette flow with closing plates attached to the outer cylinder is an imperfect model for accretion disk flows, especially with regard to their stability.
eurm10 \checkfontmsam10 Azimuthal velocities and axial transport in Rayleigh-stable Taylor-Couette]Azimuthal velocity profiles in Rayleigh-stable Taylor-Couette flow and implied axial angular momentum transport F. Nordsiek et al.]Freja Nordsiek,\nsSander G. Huisman,\nsRoeland C. A. van der Veen,\nsChao Sun ††thanks: Email address for correspondence: firstname.lastname@example.org,\nsDetlef Lohse††thanks: Email address for correspondence: email@example.com,\nsand Daniel P. Lathrop††thanks: Email address for correspondence: firstname.lastname@example.org 2010 \volume650 \pagerange119–126
Rotating shear flows are common in nature. Geophysical and astrophysical examples include the interiors of planets and stars, planetary atmospheres, and stellar and planetary accretion disks. Since direct observations and measurements are hard to perform for many of these flows, laboratory models that incorporate the essential features of these flows can be useful. A common simple rotating shear flow that can be implemented in the laboratory is Taylor-Couette (TC) flow, which is the flow in the fluid-filled gap between two coaxial rotating cylinders. Taylor-Couette flow has found particular applicability as a model for astrophysical accretion disks in determining their stability properties and the outward angular momentum flux which is necessary in order for material to be transported inward onto the central body (Zeldovich, 1981; Richard & Zahn, 1999; Richard, 2001; Dubrulle et al., 2005a; Ji & Balbus, 2013). Taylor-Couette experiments have produced contradictory answers to these questions, causing great debate centered on the effects of the no-slip axial boundaries found in Taylor-Couette experiments which do not match the open stratified boundaries of accretion disks Balbus (2011); Avila (2012); Schartman et al. (2012); Ji & Balbus (2013); Edlund & Ji (2014).
We can define a Reynolds number for the inner (outer) cylinder using the radius (), the rotation rate (), and the fluid’s kinematic viscosity , giving
Rather than using and , we use the shear Reynolds number and the so so-called parameter, detailed below, to compare different parts of the parameter space. They have a more intuitive relation to the shear and the global rotation. The shear Reynolds number , which quantifies shear, is defined as
where is the radius ratio (Dubrulle et al., 2005a). Next to , another important geometric quantity is the aspect-ratio , which is the ratio of the height of the cylinders to the gap width. To quantify the global rotation, we use the parameter (Ji et al., 2006; Schartman et al., 2012) defined through the relation
The parameter is real for co-rotating cylinders, the case exclusively dealt with in this paper. Hence, we will define both and to be both positive throughout this paper. Solid-body rotation corresponds to , gives , gives , and pure inner and pure outer rotation correspond to and respectively.
Different dimensionless parameters other than and have been used, which are presented here for ease of comparison. Rather than using to quantify the shear, previous work on our apparatus (van Gils et al., 2011b, 2012) has used a Taylor number where is a geometric Prandtl number (Eckhardt, Grossmann & Lohse, 2007), which equals for our . With a nearly identical geometry, Paoletti & Lathrop (2011) used a different definition of the Reynolds number, namely . Another parameter quantifying global rotation is the rotation parameter, (Dubrulle et al., 2005a) defined as .
At low , before the formation of Taylor-vortices, and in the absence of Ekman pumping from axial boundaries (e.g. periodic or free-slip axial boundary conditions); the azimuthal velocity profile is
We will refer to this as laminar Taylor-Couette flow.
Some rotating flows have radially increasing specific angular momentum (), where is the specific angular momentum and is the fluid angular velocity. Such flows (see figure 1); as long as they are purely hydrodynamic, barotropic, and stably stratified as we consider here; are stable to infinitesimal perturbations (i.e. linearly stable) according to the Rayleigh criterion (Rayleigh, 1917). For Taylor-Couette flow, this corresponds to . Flows for which are linearly unstable at sufficiently high Reynolds numbers (Taylor, 1923), which is often called the centrifugal instability. Hence is referred to as the Rayleigh line. The Rayleigh-stable region includes sub-rotation (), solid-body rotation (), and super-rotation (). The flow in the super-rotating region is often referred to as quasi-Keplerian, since it includes cylinder rotation rates () obeying Kepler’s 3rd law relating orbital radius and period. This regime is of particular relevance to astrophysical systems such as accretion disks since they are Rayleigh-stable with azimuthal flow profiles in the plane of the disk that are expected to not deviate significantly from Kepler’s 3rd law when ignoring the disk’s self-gravitation and relativistic effects (Richard & Zahn, 1999; Richard, 2001; Dubrulle et al., 2005a; Ji & Balbus, 2013).
Accretion disks are Rayleigh-stable but are known to have accretion rates requiring radial fluxes of angular momentum far greater than the flux provided by viscous diffusion in laminar Taylor-Couette flow-like disks, indicating that they are in fact unstable (Richard & Zahn, 1999; Richard, 2001; Dubrulle et al., 2005a; Ji & Balbus, 2013). There has been a search for the instabilities at play in these flows. Disks sufficiently ionized to be electrically conductive are known to be unstable via the Magneto Rotational Instability (MRI) (Ji & Balbus, 2013, and description therein). For weakly ionized disks or parts of disks, investigation has focused on stability in the presence of stratification (Dubrulle et al., 2005b; Le Bars & Le Gal, 2007; Le Dizès & Riedinger, 2010) and stability to finite amplitude perturbations (non-linear stability) which has been the subject of several Taylor-Couette experiments including our own in this paper (Richard, 2001; Ji et al., 2006; Paoletti & Lathrop, 2011; Paoletti et al., 2012; Schartman et al., 2012; Edlund & Ji, 2014).
For an incompressible fluid in the Rayleigh-stable region of Taylor-Couette flow and compressible accretion disk flow, the possibility of a non-linear instability has not yet been ruled out for all . Plane Couette flow and pipe flow are both examples of linearly stable flows that have non-linear instabilities at sufficient (Grossmann, 2000; Avila et al., 2011; Shi et al., 2013, and references therein). Maretzke et al. (2014) found transient growth, a necessary prerequisite for a non-linear instability, in Rayleigh-stable Taylor-Couette flow. Accretion disks have very high Reynolds numbers with possibly as high as (Paoletti et al., 2012; Ji & Balbus, 2013). Therefore, it is reasonable to ask whether Rayleigh-stable Taylor-Couette flow is non-linearly stable or unstable.
In prior experimental work; visualization via Kalliroscope particles, angular momentum transport measurements, and velocimetry measurements were done; yielding contradictory results on the presence of a non-linear instability, especially for quasi-Keplerian flow (Wendt, 1933; Taylor, 1936a, b; Coles, 1965; Richard, 2001; Ji et al., 2006; Borrero-Echeverry et al., 2010; Paoletti & Lathrop, 2011; Burin & Czarnocki, 2012; Schartman et al., 2012; Paoletti et al., 2012; Edlund & Ji, 2014). These experiments have, to varying degree, Ekman pumping driven by the no-slip boundary conditions on the axial boundaries. The Ekman pumping could destabilize the flow depending on the axial end configuration in a way that would not be found in astrophysical accretion disks (Balbus, 2011; Avila, 2012; Schartman et al., 2012; Ji & Balbus, 2013; Edlund & Ji, 2014), which have open stratified axial boundaries. Axial boundaries that rotate with the outer cylinder, such as those on the apparatus presented in this paper, were found to have Ekman pumping effects that spanned the whole flow volume (Avila, 2012; Schartman et al., 2012; Edlund & Ji, 2014), which might explain the large, and likely turbulent, angular momentum transport found by the Maryland experiment (Paoletti & Lathrop, 2011) in contrast to the low angular momentum transport steady laminar flow found in the Princeton MRI and HTX experiments which reduced the Ekman pumping by splitting the axial boundaries into rings rotated at speeds intermediate that of the two cylinders (Ji et al., 2006; Schartman et al., 2012; Edlund & Ji, 2014).
The effect of the Ekman pumping in wide-gap () low aspect-ratio () Rayleigh-stable experiments, such as the Princeton MRI and HTX experiments, on the flow state and angular momentum transport has been the subject of several investigations. When the axial boundaries are attached to the outer cylinder as opposed to rotating at intermediate speeds, there are large fluctuations and mixing near the inner cylinder (Dunst, 1972; Edlund & Ji, 2014) and quiescent flow rotating close to near the outer cylinder (Dunst, 1972; Kageyama et al., 2004; Schartman et al., 2012; Edlund & Ji, 2014). Speeding up the part of the axial boundaries near the inner cylinder causes the fluctuations near the inner cylinder to decrease and the azimuthal velocities to more closely match laminar Taylor-Couette flow (Edlund & Ji, 2014). In the reduced Ekman pumping configuration, perturbations by jets from the inner cylinder were found to decay for (Edlund & Ji, 2014).
The effects of the Ekman pumping in medium-gap () larger aspect-ratio () experiments, such as the Maryland and our experiments, has not received as much attention, though it has been expected to be similar, which would resolve the contradictory results. At low , axial boundaries attached to the outer cylinder were found to destabilize the flow (Avila, 2012). For Direct Numerical Simulations (DNS) with periodic axial boundaries in the quasi-Keplerian regime, Ostilla-Mónico et al. (2014) found that initial turbulent states always decayed to laminar Taylor-Couette flow. In this paper, we present azimuthal velocimetry profiles in both the quasi-Keplerian and the sub-rotating regimes in a geometry similar to the Maryland experiment. We compare them to the profiles in laminar Taylor-Couette flow and discuss their structure to better elucidate the effects of Ekman pumping on the flow and the large angular momentum transport associated with axial boundaries fixed to the outer cylinder for our geometry (Paoletti & Lathrop, 2011).
The paper is organized as follows: section 2 describes the experiment and the parameter space explored, section 3 presents the azimuthal velocity and specific angular momentum profiles, section 4 presents further analysis and discussion of the azimuthal profiles including the primarily axial transport of angular momentum, and section 5 summarizes the results and presents conclusions.
2 Experiment and explored parameter space
The apparatus is described in detail in van Gils et al. (2011a) and is shown schematically in figure 2. A brief summary is given here. The inner cylinder has an outer radius of cm and the transparent outer cylinder has an inner radius of cm, which gives . They can independently rotate up to a maximum of Hz and Hz, respectively. The total height is cm, which gives . The axial boundaries are attached to the outer cylinder. The inner cylinder is split into three sections. The height of the middle section is cm, and the end sections have equal heights of cm. There is a mm gap between each section, labeled 1 and 2 in figure 2. The system was filled with water and operated at room temperature with cooling applied at the axial boundaries. This geometry is similar to the apparatus used by Paoletti & Lathrop (2011), who have a and geometry.
The azimuthal velocity profiles were obtained using Laser Doppler Anemometry (LDA). The LDA configuration used backscatter from seed particles in a measurement volume of approximately mm mm mm. Dantec PSP-5 particles with a m diameter and density were used. The optical effect of the outer cylinder curvature on the LDA measurements was corrected by using the calculations of Huisman et al. (2012b). The velocimetry was calibrated using radial and axial profiles of solid-body rotation at different rotation rates. The error in the mean velocity profiles from the calibration, which was the dominant source in the mean profiles, was smaller than . For all LDA measurements a statistical convergence of was achieved, which translates to between and of , which prevents investigation into fluctuations and deviations from axisymmetry. When measuring close to the inner cylinder, reflections from the metal inner cylinder were found to be problematic. Hence, the radial profiles presented in this paper were done at the axial height of the mm gap between the bottom and middle inner cylinder sections, which corresponds to an axial height off the bottom, so that the LDA laser would be absorbed in the gap as opposed to being reflected off the cylinder surface. The axial dependence of the angular velocity was found to be less than of from axial profiles at midgap from midheight to cm off the bottom, and between radial profiles over the outer half of the gap at five heights m off the bottom, which are at . The last one, is 5 mm from the top axial boundary. Thus, a radial profile at is representative, other than possibly for radial positions closer than mm to the inner cylinder. The boundary layers on the axial boundaries are confined to within 5 mm of the boundaries.
Velocimetry was performed for five quasi-Keplerian values including Keplerian (), three sub-rotating values of , and one unstable but very close to the Rayleigh line value (); which are all listed in table 1. The value was chosen to match the simulations of Avila (2012) on a nearly identical geometry and the Princeton experimental work at (Ji et al., 2006; Schartman et al., 2012). Also, were chosen to match ongoing torque measurements on the Maryland experiment. Measurements for all values of were taken at , the three values of at , and at . All of the azimuthal velocity profiles, radial profiles at all 5 heights and the axial profile at midgap, are available in the supplementary material. Each pair of and was reached by starting with both cylinders at rest, linearly increasing and to their final values over s while maintaining constant , and then waiting at least s for transients to decay before doing measurements.
3 Results on the azimuthal profiles
It is convenient to look at the velocity profiles in terms of the normalized radial position and the normalized angular velocity given by
where is the width of the gap. The expression for gives at the inner cylinder and at the outer cylinder. Regardless of which cylinder has the larger angular velocity, the expression for gives whenever and whenever . For the quasi-Keplerian regime, indicates a super-rotating flow with , and indicates a sub-rotating flow with . Due to the sign change in the denominator for the sub-rotating regime, indicates a sub-rotating flow with and implies a super-rotating flow with . The laminar Taylor-Couette profile, which in these normalized variables is independent of and , is
The profiles for all values of at are compared to each other and to the laminar Taylor-Couette profile in figure 3. None of the profiles matched the laminar Taylor-Couette profile. Approaching solid-body rotation () at fixed in both regimes, deviation from the laminar Taylor-Couette profile increased and the part of the profile near the inner cylinder steepened. For the quasi-Keplerian regime, as we approach solid-body rotation, the rest of the profile flattens towards . For the sub-rotating regime, away from the inner cylinder. This indicates that the fluid is super-rotating in terms of angular velocity compared to both cylinders () with the degree of super-rotation, as a fraction of , increasing as we approach solid-body rotation. This flow super-rotation will be further discussed in section 4.1.
The resulting profiles of the specific angular momentum at are shown in figure 4. For the quasi-Keplerian regime, the specific angular momentum profiles all follow the same pattern of having an inner flat region connected to an outer region rotating at , which will be discussed further in section 4.2. The flat region in indicates that the flow was well mixed in that region.
Keplerian () profiles for three different are compared in figure 5. They all have a similar shape; but as is increased, decreases towards solid-body rotation at , especially in the outer parts of the gap. In terms of the specific angular momentum, increasing leads to a sharper transition between the flat region and the rotation at region.
4 Further analysis and discussion
4.1 Super-rotating flow for the sub-rotating regime
As seen in figure 3 for all three sub-rotating profiles, except near the inner cylinder indicating flow super-rotation (figure (b)b). The flow super-rotation can be quantified by taking the minimum in the profile to be the strength of the super-rotation, and finding its radial position along with where the linear interpolation of where the profile crosses () to super-rotation. The strength of the super-rotation is shown in figure (a)a, and the radial locations of the maximum super-rotation and of are shown in figure (b)b. Approaching solid-body rotation () at fixed non-zero , the strength of super-rotation increases, and the radial positions of the super-rotation maximum and of both move towards the inner cylinder. This is a singular limit, which is very different from the limit in which case one would get . The distance between these radial positions was approximately the same for all three , namely a value of gap-widths. The flow super-rotation was seen at all five heights for which radial profiles of the velocity were taken. They vary from each other by axially over the outer half of the gap. The gap-width separation was seen at the other heights for , but could not be resolved for since the point where lies in the inner half of the gap.
The specific angular momentum profiles in figure 4 were slightly greater than for solid-body rotation with the outer cylinder, except close to the inner cylinder, which is another way of saying there is flow super-rotation. The Navier-Stokes equation does not constrain angular velocities to be bound by and due to its non-linear term, unlike the temperature field in Rayleigh-Bénard flow, which is contrained between the two plate temperatures as the temperature advection equation is linear. Even with the super-rotation, we still have over the parts of the gap that are resolved; and is bound between the specific angular momenta of the outer cylinder and the axial boundaries at , which are the locations of the largest and smallest on the axial boundaries, respectively. Angular momentum is transported to the inner cylinder in this regime since the torque on the inner cylinder is negative (Paoletti & Lathrop, 2011). With inward advection of angular momentum across the gap (there is also the possibility of axial transport), the outer cylinder and axial boundaries must be the source of angular momentum to sustain the flow super-rotation against spin down to . This also allows one to estimate the maximum flow super-rotation that could be seen. If fluid from the outer cylinder having specific angular momentum is transported to the inner cylinder while conserving , it will have an angular velocity . Normalizing the flow super-rotation respectively by , , and , we get
as estimates of the super-rotation upper bound. For our , . The flow super-rotations we see in figure (a)a are one to two orders of magnitude smaller than the estimated bounds. As equation (8) diverges as , an open question is whether the magnitude of the flow super-rotation normalized by diverges as at fixed non-zero .
4.2 Quasi-Keplerian angular momentum profile and transport
For all the quasi-Keplerian profiles in figure 4, there is a pattern in the profiles. Namely, they are split into three regions: an inner region whose angular momentum profile is nearly flat with a slight positive slope, an outer region where the flow is nearly in solid-body rotation at , and a middle transition region in which the angular momentum profile curves upward from being flat to solid-body rotation at . At , the inner region extends over nearly the whole gap. As decreases for fixed , the inner region shrinks until for it is nearly absent, with the outer region having grown to be almost the whole gap.
As seen in figure (b)b, as is increased, the inner and outer regions appear to grow while the middle region shrinks. The same pattern is seen going from to for , which is not shown here but can be seen in the data in the supplementary material. The pattern suggests that in the limit , the middle region might disappear entirely. If we approximate the inner region as a completely flat angular momentum profile, approximate the outer region as rotating at exactly , ignore any boundary layer on the inner cylinder, and assume that the pattern holds for the rest of the quasi-Keplerian regime and that no flow state transitions at higher break it; then the angular velocity profile for the quasi-Keplerian in the asymptotic limit regime in our geometry would be
where is the transition radius between the flat angular momentum profile and solid-body rotation at . For large but finite , equation (11) can serve as an approximate profile. This approximate profile was derived independently by Dunst (1972) by assuming that the inner region had a flat angular momentum profile, based on his observation of a well-mixed inner region in his Taylor-Couette experiment.
For the approximately constant specific angular momentum inner region, . Then for the outer regions rotating at approximately , . Hence, we can quantify the radial position of the transition region by finding the radial positions for which the azimuthal velocity profiles are at their minimum. They are shown in figure 7. In the quasi-Keplerian regime, we find that the position of the minimum velocity corresponds very well with in equation (13), giving merit to the approximate profiles of equation (11). Outside of the quasi-Keplerian regime, the position of the minimum is located at the inner cylinder for , and at the outer cylinder for .
The approximately flat angular momentum profile in the inner region, when away from the Rayleigh line where the laminar Taylor-Couette profile is flat, indicates that the angular momentum is well mixed with advection-dominated transport in the radial direction. In contrast, there is likely little radial angular momentum transport by advection or diffusion in the outer region as the profile is close to solid-body at . A large amount of angular momentum is transported radially from the inner cylinder based on the torque measurements with the similar Maryland experiment (Paoletti & Lathrop, 2011) and on the upcoming analysis of section 4.3. The large amount of angular momentum transported off the inner cylinder and mixed in the inner region has to go somewhere, but the outer region, if present, is likely not transporting much angular momentum. Then, when an outer region is present such as when far from the Rayleigh line, most of the angular momentum must be transported axially to the axial boundaries in the inner and possibly middle regions, as shown schematically in figure 8. As increases at fixed towards the Rayleigh line, the outer region disappears and an increasing fraction of the angular momentum can be transported to the outer cylinder through the middle region instead of being transported to the axial boundaries. For , there is no middle region and a boundary layer forms close to the outer cylinder that steepens with increasing (van Gils et al., 2012), indicating that an increasing fraction of the angular momentum is transported to the outer cylinder instead of to the axial boundaries. Finally, nearly all of the angular momentum is transported to the outer cylinder.
These features are also seen in wide-gap low aspect-ratio experiments. Using dye injection from the inner cylinder, Dunst (1972) found a well mixed inner region and a quiescent outer region with poor mixing. In figure 9, the angular velocity and the specific angular momentum profiles for from Schartman et al. (2012), Edlund & Ji (2014), and Kageyama et al. (2004) are compared to each other and to the results from our apparatus. They all deviate from the laminar Taylor-Couette profile and show the same three regions with a relatively flat close to the inner cylinder and rotate close to close to the outer cylinder. However, the relatively flat inner region is offset downward from the specific angular momentum on the inner cylinder, indicating the presence of a boundary layer on the inner cylinder more significant than in our experiment. The experiments of Kageyama et al. (2004) and possibly Edlund & Ji (2014) also exhibit flow sub-rotation () in the middle and outer regions. The axial transport of angular momentum and the presence of three regions in the quasi-Keplerian azimuthal velocity profiles appear to be more general than just occuring in our specific apparatus with its geometry and ranges of and , although the strength of the boundary layer on the inner cylinder appears to depend on and/or .
4.3 Torque on the inner cylinder
The velocity gradients near the inner cylinder were larger than in laminar Taylor-Couette as the values at the point closest to the inner cylinder in figures 3 and (a)a are below that of the laminar Taylor-Couette profile. This steepness means that the torque on the inner cylinder must be larger than in laminar Taylor-Couette flow. If boundary layers were present, the profiles would be even steeper at the inner cylinder, and thus the torques even larger.
The azimuthal shear stress, when averaged azimuthally, is where is the fluid density (see page 48, Landau & Lifshitz, 1987). The torque on a cylinder of radius from just the shear stress is , which in terms of the angular velocity is
As laminar Taylor-Couette flow has no Reynolds stresses and is uniform over a cylinder of radius from equation (4), the total laminar Taylor-Couette torque is
Assuming a turbulent boundary layer, the thickness of the viscous sublayer on the inner cylinder is where is the friction velocity, is the fluid density, and is the sublayer thickness in dimensionless units (Schlichting, 1979). From measurements in our apparatus for pure inner cylinder rotation at comparable , is in the range of – (Huisman et al., 2013). Then for , we get mm since and . Since mm was the point closest to the inner cylinder where the flow velocity was resolved, our azimuthal velocimetry did not extend into the viscous sublayer. Due to not resolving the viscous sublayer, the torque in our apparatus cannot be obtained from the velocity profiles; meaning direct comparisons cannot be done to the torque measurements of Paoletti & Lathrop (2011) on the Maryland experiment with near identical geometry. However, lower bounds on the torque can be obtained because the azimuthal profiles can give the shear stress, instead of both the shear and Reynolds stresses.
To get the lower bound for the torque on the inner cylinder, was obtained from the difference between at the point closest to the inner cylinder ( mm which is ) and at the inner cylinder. It must be noted that the velocity profile was taken at the axial height of one of the small separations in the inner cylinder, which is mm thick, and therefore the gradients in we calculate might be perturbed compared to other axial heights due to the vicinity to the separation.
The torque lower bounds are listed in table 2. The lower bounds were all larger than the laminar Taylor-Couette torque, which supports the result of Paoletti & Lathrop (2011) on the similar Maryland experiment in both regimes for . The measurements in this paper extend this result of Paoletti & Lathrop (2011) towards solid-body rotation in both regimes.
For the quasi-Keplerian regime, we can use the approximate flatness of the specific angular momentum profile in the inner region to make an analytical approximate torque lower bound. Treating the inner region as having a flat specific angular momentum profile from the inner cylinder with no boundary layer as in equation (11), the ratio of the torque lower bound to the laminar Taylor-Couette torque is
The ratio is always larger than one, approaching one at . It diverges as , which is due to the width of the inner region shrinking towards zero since in equation (12). The decrease in means that changes from to over an ever smaller radial distance, giving a sharper gradient of in the inner region, which becomes infinite as . However, if the inner region of a flat angular momentum profile disappears entirely as at a given , then this lower bound may no longer hold. For , the inner region might be close to disappearing by based on the angular momentum profiles in figure 4. As the middle region shrinks with increasing (figure (b)b), the at which the inner region might disappear decreases with increasing .
The torque lower bounds can be compared to the torque scaling that Paoletti et al. (2012) fit to the Maryland torque measurements in the Rayleigh-stable and unstable regimes (Paoletti & Lathrop, 2011) and the torque measurements on the apparatus presented in this paper in the unstable regime (van Gils et al., 2011b). The scaling was for the ratio of the torque on the inner cylinder to the torque for pure inner rotation () at the same , which in this paper was obtained from torque measurements in the very similar Maryland experiment (equation (9) in Lathrop et al., 1992). The lower bounds are compared to the torque scaling (equation (12) in Paoletti et al., 2012) in figure 10.
As the torque ratios must be positive, the torque scaling must start curving upwards on the sub-rotating regime side when approaching solid-body rotation at some to avoid crossing zero. The three sub-rotating regime torque lower bounds for give values that are larger than those for (Paoletti et al., 2012). Thus, the scaling of Paoletti et al. (2012) must increase if extended to .
On the quasi-Keplerian side, comparisons can be made between our measurements at to those of Paoletti et al. (2012) for . Our lower bounds from both the measured velocity profiles and the flat inner region approximation from equation (16), are considerably smaller than those of Paoletti et al. (2012). As our lower bounds only considered shear stress (diffusion), the difference in torques on the inner cylinder must be due to Reynolds stresses (advection) in the region of . The divergence of the torque lower bound for the flat inner region angular momentum approximation as suggests that the flat quasi-Keplerian scaling of Paoletti et al. (2012) will deviate from being flat if extended to , unless the inner region disappears or is distorted close to solid-body rotation.
5 Summary and conclusions
In summary, azimuthal velocity profiles were obtained for several Rayleigh-stable (and one unstable) cylinder rotation rate ratios for the ranges and . They were all done for , a few configurations at , and just the Keplerian configuration also at . For all values of , the profiles deviate from the laminar Taylor-Couette profile. The deviation increases as solid-body rotation is approached () at fixed non-zero . The deviation consists of a steepening of the normalized angular velocity profile close to the inner cylinder for all , and the flow in the outer parts of the gap approaching solid-body rotation with the outer cylinder and attached axial boundaries for the quasi-Keplerian regime.
For the sub-rotating regime, the flow exhibits super-rotation compared to both cylinders (), except close to the inner cylinder. As solid-body rotation is approached at fixed , the strength of the super-rotation increases, reaching of for , and the radial positions of the maximum of super-rotation and where the flow switched from to super-rotation moves closer to the inner cylinder. The flow super-rotation must be sustained by inward angular momentum transport from the outer cylinder or axial boundaries. To the best of our knowledge, flow super-rotation for has not been previously observed in the literature. This includes pure outer-rotation () in our apparatus (van Gils et al., 2011a) and in those of Taylor (1936b), Wendt (1933), and Burin & Czarnocki (2012).
For the quasi-Keplerian regime, the specific angular momentum profiles show that the flow can be split into three regions across the gap: an inner region where the angular momentum profile is approximately flat, an outer region where the flow is close to solid-body rotation at , and a middle transition region between the two. Starting near the Rayleigh line, the middle and outer regions are almost non-existent; and then as solid-body rotation is approached at fixed non-zero , the inner region shrinks while the outer region grows till the inner region is almost non-existent at . As is increased, the middle region shrinks. We speculate that as , the middle region will disappear and the profile will converge towards equation (11) (independently derived from dye injection observations by Dunst, 1972). This model profile is a good approximation by . The outer region, if present, likely transports little angular momentum, meaning that almost all of the angular momentum is transported to the axial boundaries. Work is still needed to check how the mixing is achieved in the inner region and whether the inner region is turbulent. One must also determine the exact nature of the flow in the middle and outer regions, and see whether the region pattern is found for other and . Measuring the separate torques on the inner cylinder, outer cylinder, and axial boundaries or doing very high resolution Particle Image Velocimetry as Huisman et al. (2012a) did for pure inner rotation on the same apparatus would be good ways to determine what fraction of the angular momentum goes to the axial boundaries versus the outer cylinder and elucidate the axial transport mechanism. We do not see flow sub-rotation in our experiment except possibly for and , but within the measurement precision at those . However, Kageyama et al. (2004) and possibly Edlund & Ji (2014) found sub-rotating flow for .
The slope of the angular velocity profile at the inner cylinder is steeper than in laminar Taylor-Couette flow for both the sub-rotating and the quasi-Keplerian regimes. Therefore, the torque required to rotate the inner cylinder must be larger than the laminar Taylor-Couette value, which supports the super-laminar torque measurements on the geometrically similar Maryland experiment (Paoletti & Lathrop, 2011). Due to not resolving the viscous sublayer on the inner cylinder, only lower bounds for the torque on the inner cylinder could be obtained via a viscous stress calculation. The lower bounds from the velocity measurements and the approximate asymptotic profile were compared to the scaling found by Paoletti et al. (2012) for . In the sub-rotating regime, their scaling needs to be increased if extended to or towards solid-body rotation. In the quasi-Keplerian regime, the comparison also shows that the bulk of the transport of angular momentum off the inner cylinder is by Reynolds stresses (advection) and the scaling of Paoletti et al. (2012) may require modification as .
Our velocity profiles provide experimental confirmation of the expectation that the Ekman pumping from the axial boundaries was what destabilized the flow in the Maryland experiment, which has a nearly identical geometry as our apparatus, in the Rayleigh-stable regime causing large super-laminar torques on the inner cylinder (Balbus, 2011; Avila, 2012; Ji & Balbus, 2013; Edlund & Ji, 2014). This work, combined with the work of Avila (2012), Schartman et al. (2012), and Edlund & Ji (2014), resolves the apparent discrepancy between the approximately laminar Taylor-Couette angular momentum transport in the wide-gap, low aspect-ratio experiments with axial boundaries split into rings rotating at speeds intermediate that of the cylinders such as the Princeton MRI and HTX experiments (Ji et al., 2006; Schartman et al., 2012; Edlund & Ji, 2014) and large super-laminar angular momentum transport in the medium-gap higher aspect-Maryland experiment with axial boundaries attached to the outer cylinder (Paoletti & Lathrop, 2011). Moreover, we found that the Ekman pumping from the axial boundaries does more than just destabilize the flow in the Rayleigh-stable regime when the axial boundaries are attached to the outer cylinder. In the quasi-Keplerian regime, it causes the flow to be split radially into three regions and nearly all of the angular momentum to be transported to the axial boundaries instead of the outer cylinder when an outer region is present. The Ekman pumping essentially causes the axial boundaries to become the primary sink of angular momentum. In the sub-rotating regime, we discovered flow super-rotation, which is also likely due to the Ekman pumping.
Astrophysical accretion disks have open axial boundaries, which do not cause Ekman pumping, and are thought or assumed to have primarily radial transport of angular momentum (Zeldovich, 1981; Richard & Zahn, 1999; Richard, 2001; Dubrulle et al., 2005a; Ji & Balbus, 2013; Ostilla-Mónico et al., 2014). Due to the strong Ekman pumping effects, including the primarily axial transport of angular momentum, Taylor-Couette flow with an aspect ratio up to with no-slip axial boundaries attached to the outer cylinder is an imperfect model of accretion disks, especially with regard to stability. Ideally, one would like to have axial boundaries that are free-slip or rotate at different rates along their radius such that they match the mean rotation rate of what the flow would be in the absence of axial boundaries, which may not be the laminar Taylor-Couette profile.
There are practical options available to experimental Taylor-Couette flow to mitigate the Ekman pumping and make a better model of accretion disks. One practical way is to make an experiment where the aspect ratio is great enough that the axial tranport mechanism of the angular momentum saturates and Ekman pumping can no longer directly affect the flow near midheight. However, tall experiments are difficult to handle and expensive to make, and work would be needed to ascertain whether indirect effects would still be a problem. Another way, which has been followed by the Princeton group (Ji et al., 2006; Schartman et al., 2012; Edlund & Ji, 2014), is to split the axial boundaries into rings that are rotated at speeds intermediate to those of the cylinders. This reduces the strength of the Ekman pumping as well as better confining it to the axial boundaries. Implementing the independently rotating rings is difficult and there is still Ekman pumping due to having only a finite number of independently rotating rings. If the working fluid is a liquid, the top boundary can be made into an open boundary by having gas above it, reducing the Ekman pumping at the top by three orders of magnitude in the case of water and air, though it does introduce the problem of gravity waves on the top surface. For the velocities that are used in the present experiments, air could be entrained by these waves. Similarly, density-mismatched fluids such as mercury and water or stratification (e.g. salt solutions) can be used on the bottom boundary to confine the Ekman circulation near the bottom by reducing axial circulation, although this also introduces the problem of gravity waves and mixing which would destroy the stratification. In order to accurately represent an accretion disk, one probably has to combine more than one of these methods.
We would like to acknowledge helpful discussions and advice from Dennis P.M. van Gils, Siegfried Grossmann, Rodolfo Ostilla-Mónico, Daniel S. Zimmerman, Eric M. Edlund, and Hantao Ji. We thank Hantao Ji for providing us the velocimetry data from Kageyama et al. (2004). We also acknowledge work and advice from the technicians Gert-Wim Bruggert, Martin Bos, and Bas Benschop; and financial support from the Technology Foundation STW of The Netherlands from an ERC Advanced Grant and the National Science Foundation of the USA (Grant No. NSF-DMR 0906109).
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