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Abstract

We numerically study turbulent Taylor-Couette flow (TCF) between two independently rotating cylinders and the transition to rotating plane Couette flow (RPCF) in the limit of infinite radii. By using the shear Reynolds number and rotation number as dimensionless parameters, the transition from TCF to RPCF can be studied continuously without singularities. Already for radius ratios we find that the simulation results for various radius ratios and for RPCF collapse as a function of , indicating a turbulent behaviour common to both systems. We observe this agreement in the torque, mean momentum transport, mean profiles, and turbulent fluctuations. Moreover, the central profiles in TCF and RPCF for are found to conform with inviscid neutral stability. Intermittent bursts that have been observed in the outer boundary layer and have been linked to the formation of a torque maximum for counter-rotation are shown to disappear as . The corresponding torque maximum disappears as well. Instead two new maxima of different origin appear for and RPCF, a broad and a narrow one, in contrast to the results for smaller . The broad maximum at is connected with a strong vortical flow and can be reproduced by streamwise invariant simulations. The narrow maximum at only emerges with increasing and is accompanied by an efficient and correlated momentum transport by the mean flow. Since the narrow maximum is of larger amplitude for , our simulations suggest that it will dominate at even higher .

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eurm10 \checkfontmsam10 Momentum transport in Taylor-Couette flow with vanishing curvature]Momentum transport in Taylor-Couette flow with vanishing curvature H. J. Brauckmann, M. Salewski and B. Eckhardt]Hannes J. Brauckmann,\nsMatthew Salewskiand Bruno Eckhardt1 2010 \volume650 \pagerange119–126

keywords

1 Introduction

The motion of a fluid between two independently rotating concentric cylinders, denoted as Taylor-Couette flow (TCF), serves as a fundamental model system to study the influence of rotation on turbulence. The two main control parameters of the system, the rotation rates of both cylinders, can be combined into two parameters that capture the key physical processes. The differential rotation of the cylinders results in a shear gradient driving the flow, and the average of inner and outer cylinder rotation (denoted as mean system rotation) defines the magnitude of rotational influences which can either enhance or attenuate the fluid motion depending on the rotation direction. This decomposition into differential and mean rotation is depicted in figure 1(a). Depending on the system rotation, the laminar state becomes unstable due to a centrifugal instability (Rayleigh, 1917; Taylor, 1923) or without a linear instability via a subcritical transition (Wendt, 1933; Taylor, 1936; Burin & Czarnocki, 2012; Deguchi et al., 2014) similar to the transition in other wall-bounded shear flows. Beyond the laminar regime different combinations of shear and system rotation result in a multitude of distinct flow states (Coles, 1965; Andereck et al., 1986) that continue to shape the turbulence up to high cylinder speeds (Ravelet et al., 2010; Huisman et al., 2014).

The influence of the curvature of the cylinder walls can be captured by the ratio of the radii of the inner and outer cylinders, , as a third control parameter. If the radius ratio falls considerably below , the curvature influence on the fluid motion becomes important. Conversely, the system curvature becomes small when approaches while keeping the gap width constant, since the cylinder radii diverge in this limit. Then, the Taylor-Couette (TC) system approximates plane Couette flow subject to spanwise system rotation, c.f. figure 1(b), commonly called rotating plane Couette flow (RPCF). We here study the limiting process to vanishing curvature for turbulent TCF and its dependence on the system rotation. For this purpose, we performed direct numerical simulations (DNS) of TCF for seven values of ranging between and . For each radius ratio, three turbulent shear rates and various mean rotation states were realised. We complemented these TC simulations with analogous DNS of RPCF to compare directly to the curvature-free limit.

     

Figure 1: Schematic representations of the two flows. (a) TC system seen from above in the reference frame rotating with . Only one third of the azimuthal circumference is shown, reflecting the actual computational domain for . (b) RPCF, as obtained from (a) for vanishing curvature, i.e. when for a constant gap width . While the spanwise -coordinate points out of the paper in both systems, the wall-normal coordinates and are antiparallel.

An important physical quantity in turbulent TCF is the torque needed to drive the cylinders, which is connected to a radial transport of angular momentum by the fluid motion (Marcus, 1984; Eckhardt et al., 2000; Dubrulle & Hersant, 2002). In the turbulent regime, the torque can be characterised by its scaling with the shear strength modulated by a function that depends on the system rotation (Dubrulle et al., 2005). This rotation-dependence of the torque features a maximum for counter-rotating cylinders as measured and calculated for (Paoletti & Lathrop, 2011; van Gils et al., 2011; Brauckmann & Eckhardt, 2013a; Ostilla et al., 2013) and for (Merbold et al., 2013). The torque-maximizing rotation state has been linked to the occurrence of intermittent turbulence near the outer cylinder that decreases the momentum transport efficiency (van Gils et al., 2012; Brauckmann & Eckhardt, 2013b) and results from the stabilisation of the outer flow region for counter-rotating cylinders. At the same time, the flow near the inner cylinder remains centrifugally unstable. Since the curvature that causes this difference in stability between the inner and outer region vanishes for , we also expect the rotation influence on the torque to change. Evidence for such deviations in behaviour between cases with and larger have also been noted in experiments with (Ravelet et al., 2010; Ostilla-Mónico et al., 2014a). However, the rotation-dependence in TC systems with a radius ratio significantly larger than has not been explored, yet, and represents one of the main objectives of the present study. Moreover, we will study how turbulent characteristics, such as torques, mean profiles, and fluctuations, change during the transition from low-curvature TCF to RPCF and how TCF turns into RPCF.

In the context of the transition to turbulence, the RPCF-limit of the TC system has already been studied in an approximation where certain terms in the equations of motion have been neglected (Nagata, 1986, 1990; Demay et al., 1992). Nagata found that wavy vortices, which develop as an instability of streamwise invariant Taylor vortices, can be continued to non-rotating plane Couette flow where they form finite-amplitude solutions. Such wavy roll cells were also observed experimentally in RPCF at low shear rates (Hiwatashi et al., 2007). Moreover, by considering the full equations of motion, Faisst & Eckhardt (2000) were able to continuously study the transition from TC to plane Couette flow and found that already for radius ratios below some characteristics of plane Couette flow can be observed. Furthermore, Dubrulle et al. (2005) noted that also the linear stability criterion for the laminar state in RPCF (Lezius & Johnston, 1976) is approximated by the stability boundary of TCF (Esser & Grossmann, 1996) when the radius ratio tends to . Beyond the first bifurcation towards turbulence, further transitions create a rich set of flow states also in RPCF (Tsukahara et al., 2010; Daly et al., 2014), reminiscent of the states described by Andereck et al. (1986) in TCF. In an extension of these bifurcation studies at low Reynolds numbers that revealed a number of similarities between the two systems, we here investigate the continuous transition of TCF to RPCF with a focus on turbulent characteristics at high shear rates.

In the turbulent regime, the effect of rotation depends on the relative orientation of rotation and vorticity in plane Couette flow. Cyclonic rotation, i.e. when the rotation vector is parallel to the vorticity of the laminar base flow, was found to have a stabilising effect on the turbulence (Komminaho et al., 1996) and to generate a striped pattern of coexisting laminar and turbulent regions (Tsukahara, 2011). Similar turbulent-laminar patterns were also observed at much higher shear rates by increasing the cyclonic rotation (Brethouwer et al., 2012). On the contrary, anticyclonic spanwise rotation, where the rotation is antiparallel to the base-flow vorticity, has a destabilizing effect and was found to drive vortices reminiscent of turbulent Taylor vortices (Bech & Andersson, 1997; Barri & Andersson, 2010; Tsukahara, 2011). For strong anticyclonic rotations, these vortices break up again and become disorganised. The influence of rotation on the strength of vortices is of substantial interest also in the present study since such vortices can enhance the momentum transport and, therefore, contribute to the formation of torque maxima as demonstrated by Brauckmann & Eckhardt (2013b).

Investigations of RPCF for anticyclonic rotation furthermore revealed that the mean downstream velocity profile exhibits a linear section in the centre of the gap. In this region, which can extend over a considerable fraction of the gap, the vorticity of the mean profile compensates the imposed vorticity of the spanwise rotation (Bech & Andersson, 1997; Barri & Andersson, 2010), which means that the gradient of the central profile increases linearly with the imposed rotation rate. The arrangement of the profile to approximately realise zero absolute vorticity was recently also observed in a non-turbulent RPCF experiment, thus demonstrating the generality of this behaviour for unstable flows (Suryadi et al., 2014). A physical explanation for these central profile shapes comes from the observation that mean profiles of zero absolute vorticity comply with neutral stability in RPCF (Barri & Andersson, 2010; Suryadi et al., 2014). Such prominent changes of the profile gradient with varying system rotation were also observed for the angular velocity in TCF (Wendt, 1933; Ostilla et al., 2013; Ostilla-Mónico et al., 2014a). However, it remains unclear how the mean profiles in TCF compare to those in RPCF and whether they converge in the limit of vanishing curvature ().

The purpose of the present study is to examine the continuous transition from TCF to RPCF and to compare turbulent characteristics between both systems taking into account the following aspects. First, we quantify the effect of the vanishing curvature, when tends to unity, by the decreasing strength of intermittent fluctuations near the outer cylinder and by the local stability for counter-rotating cylinders. Second, we show that as these intermittent fluctuations disappear, the rotation-dependence of the torque also changes. Third, we look for flow characteristics that prove the convergence of TCF to RPCF and seek the range in which the convergence can be observed. Fourth, the magnitude of the momentum transport by coherent vortices is analysed to assess their importance for the formation of torque maxima. In addition, we evaluate the effect of turbulent fluctuations on the momentum transport. Finally, we look for a convergence of streamwise velocity profiles and introduce a unified description of central profile gradients in TCF and RPCF.

The paper is structured as follows. In §2 we describe the unified framework in which we study TCF and RPCF, including the equations of motion, common control parameters and their ranges, the momentum transport equations and numerical aspects. The vanishing of curvature effects is studied in §3, followed in §4 by an analysis of the total momentum transport as well as the vortex-induced momentum transport. Finally, we discuss the rotation-dependence of velocity profiles and of turbulent fluctuations contributing to the momentum transport in §5. We close with a summary and some concluding remarks.

2 Definitions and numerical methods

Since there are several different combinations of parameters in TCF, we begin by discussing the limiting case of RPCF, so that we can then chose parameters that approach the ones in RPCF. As usual, RPCF refers to the flow between two parallel walls that move in opposite directions and are subject to system rotation in the spanwise direction. We describe this system in Cartesian coordinates with the velocity field and the walls at and moving in the -direction with velocities and , respectively. The entire system rotates with angular velocity around the spanwise axis as depicted in figure 1(b). We chose and as characteristic scales for all lengths and velocities. In these units and in a reference frame rotating with , the Navier-Stokes equation and incompressibility condition become

(1)

where denotes the non-dimensionalised pressure; the centrifugal force has been absorbed into the pressure. The motion is characterised by two dimensionless parameters, the shear Reynolds number and the rotation number which measures the ratio between system rotation and shear,

(2)

where denotes the kinematic viscosity of the fluid. In contrast to the often used plane Couette flow Reynolds number , the definition of the shear Reynolds numbers is based on the full velocity difference and wall distance, so that .

TCF is the motion of a fluid between independently rotating concentric cylinders. Its geometric parameters are the inner and outer cylinder radii and and the axial height . The radius ratio, defined as , determines the curvature; it is thus the parameter that controls the transition to RPCF (which emerges for ). The inner and outer cylinders rotate with the angular velocities and , respectively, and their ratio or its negative, , (van Gils et al., 2011; Brauckmann & Eckhardt, 2013a; Ostilla et al., 2013) is often used to define the rotation state of the system. In order to study TCF in the same framework as RPCF, we follow the analysis of Dubrulle et al. (2005) and describe the flow in a reference frame that rotates with the angular velocity . We make use of the cylindrical symmetry and introduce coordinates with velocities . In analogy to RPCF, we choose such that the cylinder walls move symmetrically in the opposite direction in the rotating reference frame, i.e. such that the condition is fulfilled as depicted in figure 1(a). This gives

(3)

Furthermore, we choose the velocity difference between the cylinder walls as the characteristic velocity scale in the rotating reference frame, i.e.

(4)

The last step in (4) results from substituting from equation (3). With a view towards the variables in RPCF, we take the gap width as the characteristic scale to measure all lengths. Instead of the radius ratio , Dubrulle et al. (2005) specify a typical radius and define the curvature number to characterise the curvature of the system. (The relation to is given in equation (8) below, where the definitions of all parameters are collected.) Using these units, the equations of motion in cylindrical coordinates (Landau & Lifshitz, 1987) transformed to the rotating reference frame can be written as

(6)

with the modified Nabla and Laplace operators

(7)

Again the centrifugal force term is absorbed in a modified pressure . The terms in the equations (2)–(7) are arranged in a form adapted from Faisst & Eckhardt (2000) in order to clarify that in the limit the equations of motion (2) and (6) indeed converge to the corresponding ones for RPCF (1) since the additional terms on the right hand side vanish as . The transition between Cartesian coordinates in RPCF and cylindrical coordinates in TC requires the identifications and .

The equations of motion show that the TC system is characterised by three dimensionless parameters (Dubrulle et al., 2005),

(8)

that result from (3) and (4), where and denote the traditional Reynolds numbers that measure the dimensionless velocity of the inner and outer cylinders in the laboratory frame of reference. Note that and are defined in strict analogy to RPCF (2) so that has the opposite sign of the rotation number used by Dubrulle et al. (2005); Ravelet et al. (2010); Paoletti et al. (2012). Finally, in these units the laminar angular velocity profile in the laboratory frame given by the circular Couette flow reads

(9)

with ranging between and .

2.1 Parameters for the limit of vanishing curvature

We are interested in the transition from TCF to RPCF, in the limit when the radius ratio approaches one. This limit is often called the small-gap limit, since it can be obtained for fixed outer radius when the radius of the inner cylinder approaches that of the outer one and the gap width ultimately vanishes. However, in units of the width , the radii increase like and diverge as approaches one; this is why this limit was called the case of “large radii” in Faisst & Eckhardt (2000). Moreover, the curvature vanishes () as , suggesting that curvature effects become less important. However, rotation effects can be maintained.

Figure 2: Transformation between the ratio of angular velocities and the rotation number for various radius ratios . For TC systems with vanishing curvature (), most -values result in almost no system rotation () and is achieved only for close to one. The symbols mark the values of and realised by our simulations for .

For every radius ratio , the traditional Reynolds numbers can be expressed in terms of the shear Reynolds number and the rotation number as

(10)

and more specifically as and in the non-rotating case () for all . In contrast, for finite rotation () the cylinder Reynolds numbers and diverge in the vanishing-curvature limit as . As a consequence, the ratio of angular velocities,

(11)

converges to for and to for any in the vanishing-curvature limit, as shown in figure 2. These limiting values illustrate that the transition between TCF and RPCF is singular in the traditional parameters, , , and , but not in the parameters and that we will generally use here. We note that one exception to this parameter choice occurs in §3 where the effect of curvature for counter-rotating cylinders can be characterised more adequately by the ratio of angular velocities . Furthermore, while measures the curvature and thus clarifies its disappearance () for , we will identify the different TC geometries by the radius ratio as this parameter is more commonly used.

2.2 Momentum transport in Taylor-Couette and plane Couette flow

To study the mean turbulent characteristics of the flow, we calculate wall-normal profiles of the velocity and other flow fields by averaging over surfaces parallel to the wall. The averages are denoted by . The surfaces are concentric cylinders in the case of TCF, and planes parallel to the bounding walls in the case of RPCF, . Equations for the averages follow from the Navier-Stokes equation. The average of the -component of the equation of motion (2) of TCF results in the continuity equation for the specific angular momentum

(12)

where denotes the angular-momentum flux in dimensionless units (Marcus, 1984), the angular velocity and the divergence in radial direction. In the statistically stationary state, where , the transverse current becomes independent of the radius (Eckhardt et al., 2007b). In physical units, the torque needed to drive the cylinders, can be calculated from the dimensionless angular-momentum flux as , with the dimensionless cylindrical surface area and the mass density of the fluid . The remaining prefactors compensate for the non-dimensionalization of (12). We define the dimensionless torque per axial length,

(13)

and introduce the Nusselt number,

(14)

that measures the torque in units of the torque of the laminar profile (Dubrulle & Hersant, 2002; Eckhardt et al., 2007a).

Equations corresponding to (12)–(14) can also be derived for RPCF. Averaging the downstream component of the equation of motion (1) results in a continuity equation for the specific momentum in the -direction

(15)

(see also Pope, 2000, §7.1), where denotes the momentum flux in dimensionless units. It is independent of the wall-normal coordinate in the statistically stationary state. In analogy to the torque , the force needed to keep the plane walls at a constant velocity can be calculated from the momentum flux multiplied by the dimensionless surface area through which the flux passes, . Finally, we define the Nusselt number for the momentum transport in RPCF as in (14),

(16)

that measures the force in units of the value for the laminar flow.

This completes our set of relations between TCF and RPCF that we want to study here. We have identified correspondences between the continuity equations (12) and (15), the torque and the force (), the angular-momentum flux and the momentum flux (), the Nusselt numbers (), and the mean profiles that enter the diffusive part of the flux in (12) and (15) (). Since we study the statistically stationary cases only, we also average in time to improve the statistics. Using the dimensionless shear Reynolds number and rotation number (2), we can now study the momentum transport in both systems within the same framework and in a non-singular manner.

We conclude this section by pointing out a peculiar property of the relations that we will take up again in later sections. While the equations of motion (1) and (2) depend on both driving parameters and , the momentum transport equations (12) and (15) do not explicitly depend on the rotation number . Note that the same observation applies to RPCF (Salewski & Eckhardt, 2015). The rotation only indirectly influences the momentum fluxes and by changing the turbulent velocity correlations ( and ) and the mean profiles ( and ), as we will discuss for the correlations in §§4.1.1 and 5.3 and for the profiles in §5.2.

2.3 Investigated parameter range

Studies for different (or ) give information on when the properties of RPCF emerge from TCF, and studies of different and give information on the impact of shear and rotation on the turbulence. The range of parameters explored is indicated in Fig. 3. They were selected as follows.

Figure 3: Parameter space explored by DNS of TCF to study the limit of vanishing curvature, i.e. equivalent to . (a) For the shear Reynolds numbers and curvature numbers indicated here, several simulations with were performed. (b) The range of rotation numbers explored is shown here for and different curvature numbers . The curvature numbers correspond to the radius ratios , , , , , and . The extra columns for summarise our RPCF simulations.

For each curvature number the evolution of the rotation dependence is analysed with increasing shear, by realising various mean rotation states for three shear Reynolds numbers , and . Since we adopt the results for from our previous study (Brauckmann & Eckhardt, 2013a), the simulated values of , , deviate slightly from the target values, as indicated in figure 3(a). Similarly, the RPCF results adopted from Salewski & Eckhardt (2015) realise , and . In the following, we will skip over these small differences in , as they are not significant within statistical fluctuations. Our highest value of lies in the range where the transition to the fully turbulent state was observed in experiments for and stationary outer cylinder (Lathrop et al., 1992; Lewis & Swinney, 1999). We performed simulations for the range of rotation numbers where the occurrence of a torque maximum was observed for medium radius ratios (Dubrulle et al., 2005; Paoletti & Lathrop, 2011; van Gils et al., 2011; Paoletti et al., 2012; Brauckmann & Eckhardt, 2013a; Merbold et al., 2013; Ostilla et al., 2013); this allowed us to extend these studies to close to one. Moreover, rotation numbers that correspond to strong counter-rotation were selected in order to study the fate of the turbulent intermittency in the outer region (Coughlin & Marcus, 1996; van Gils et al., 2012; Brauckmann & Eckhardt, 2013b). Thus, the simulated rotation numbers lie mainly in the range as shown in figure 3(b), except for () where we focus on a narrow range in only for a study of the intermittency for counter-rotation. The case of no system rotation, i.e. , corresponds to counter-rotating cylinders with .

For the highest shear Reynolds number , the evolution of the rotation dependence is studied for seven curvature numbers (TCF) and for (RPCF), as listed in table 1. This investigation of the curvature effects compares turbulent characteristics, such as torques, mean profiles, and velocity fluctuations, between the different studied.

2.4 Numerical codes and tests of convergence

0.5 0.707 3      1 2  62 70 190 0.33 0.74 0.23
0.71 0.344 6 3.09 1 2  78 70 110 0.18 0.54 0.90
0.80 0.224 7 4.04 1 2  94 70  94 0.20 0.33 1.14
0.90 0.105 15 3.98 1 2  94 70  94 0.23 0.37 0.52
0.95 0.051 30 4.08 1 2  94 70  94 0.22 0.35 0.44
0.98 0.020 78 3.99 1 2  94 70  94 0.14 0.20 0.36
0.99 0.010 99 6.31 1 2 158 70  94 0.13 0.30 0.33
1 0  2 1 2 159 76  95 0.42 0.42 0.38
Table 1: Size of the simulations domain at the highest for the different radius ratios and corresponding curvature numbers : , and denote the streamwise (azimuthal), wall-normal (radial) and spanwise (axial) length, and , , give the highest order of spectral modes in the corresponding direction. The azimuthal periodicity was introduced for the TC simulations to reduce the natural azimuthal length to with the mean radius . The amplitudes of the highest spectral modes , and estimate the truncation error of the expansion in the corresponding direction. We listed the maximal amplitude values over all .

For our investigations of TCF and RPCF we perform DNS that approximate a solution to the incompressible Navier-Stokes equation (2)-(6) and (1), respectively. In these simulations we introduce a spanwise (axial) periodicity of length which is large enough to accommodate one pair of counter-rotating Taylor vortices. In addition, a streamwise periodicity of length was assumed in the RPCF simulations while TCF is naturally periodic in this direction. However, we only simulated a domain of reduced azimuthal length that repeats times to fill the entire cylinder circumference. For and we tested that the reduced azimuthal length does not bias the computed torques for a stationary outer cylinder. The length was further increased for larger radius ratios, see table 1. As discussed in more detail in Brauckmann & Eckhardt (2013a); Ostilla-Mónico et al. (2015), the effect of these geometrical constraints of the domain are small for single point quantities, like the torque and profiles studied here.

For the TC simulations we employ a numerical code developed by Marc Avila and described in Meseguer et al. (2007). To accomplish large rotation numbers for radius ratios close to , both cylinders have to co-rotate rapidly and reach individual Reynolds numbers and of up to while maintaining the shear . Since these high absolute velocities introduce numerical instabilities, we perform these simulation cases in a reference frame rotating with the angular velocity defined in (3). For the RPCF simulations we use Channelflow (Gibson et al., 2008; Gibson, 2012), a pseudospectral code developed by John F. Gibson that was modified to include the Coriolis force, . Both numerical codes use an expansion of the velocity field in Fourier modes in the two periodic directions and in Chebyshev polynomials in the wall-normal direction, and employ a semi-implicit scheme for the time-stepping. We denote the highest order of modes employed in the streamwise, wall-normal, and spanwise direction by , , and , respectively.

The truncation error of the expansions can be assessed by the amplitudes of the highest modes , and which are normalised by the globally strongest mode. Our resolution tests in Brauckmann & Eckhardt (2013a) revealed that these amplitudes have to drop to in order to reach a converged torque computation. All present simulations fulfil this criterion. It turned out that the resolution in the axial direction could be reduced for larger while maintaining . This seems plausible since the boundary layer for small is thinner at the inner cylinder than at the outer one (Eckhardt et al., 2007b) and thus requires a higher resolution. Nevertheless, we did not reduce the resolution in the radial direction for large . Moreover, our simulations satisfy two additional convergence criteria described by Brauckmann & Eckhardt (2013a): Agreement of the torques computed at the inner and outer cylinder to within and agreement of the energy dissipation estimated from the torque and from the volume averaged energy dissipation rate to within .

3 Curvature effects: Radial flow partitioning

The first linear instability of TCF with counter-rotating cylinders is known to develop only in an annular region close to the inner cylinder while the outer region remains stable (Taylor, 1923). Inviscid calculations predict that the boundary between these two regions is given by the neutral surface at the radius where the laminar Couette profile passes through (Chandrasekhar, 1961, pp. 276–77). However, already the observations by Taylor (1923) as well as viscous calculations suggest that the structures from the inner partition protrude beyond by a factor . Esser & Grossmann (1996) found it to be in the range between and . Beyond the first instabilities, the initially stable outer partition becomes susceptible to turbulent bursts at higher (Coughlin & Marcus, 1996) which give rise to a radial partitioning of the turbulent flow that persists for much higher : the coexistence of permanent turbulence in the inner partition and intermittent turbulence in the outer one was observed for (van Gils et al., 2012).

In Brauckmann & Eckhardt (2013b) we argued that the intermittency in the outer partition can only occur when the extended inner unstable region does not cover the entire gap. This resulted in the prediction

(17)

for the critical rotation ratio, , so that flow partitioning occurs for . The intermittent dynamics in the outer partition comes with a reduction in momentum transport and the formation of a maximum in torque as a function of . However, we already noted in our previous study that the bursting behaviour in the outer partition is a curvature effect that will not appear for and, consequently, (17) will become invalid in this limit.

Figure 4: Standard deviation (divided by the mean ) of temporal torque fluctuations calculated at the outer cylinder ( ) and inner cylinder ( ) for a constant shear and varying rotation ratio . The dotted line marks the base level for the fluctuation at the inner cylinder for strong counter-rotation (average of open triangles). The solid line is a linear fit to 6 points in (a), to 5 points in (b), to 4 points in (c) and to 3 points in (d) which are marked as open circles. The vertical dashed line indicates the intersection point that defines the onset of enhanced outer fluctuations.

In order to study what happens to the intermittency and the torque maxima, we characterize the stronger intermittent fluctuations in the outer partition by the standard deviation of the temporal torque fluctuations at the inner and outer cylinder for . This previously enabled us to identify the onset of enhanced outer fluctuations as a function of for and (Brauckmann & Eckhardt, 2013b). We here extend the analysis to and . In the previous study, the relative fluctuation amplitude was assumed to be constant (i.e. independent of ) at the inner cylinder. Therefore, the value of at served as the reference value that the outer fluctuations exceed. However, since the data for and do not show constant inner fluctuations (figure 4c,d), we here take the average inner fluctuations () for strong counter-rotation as the base level. Figure 4 shows that the outer torque fluctuations exceed this level when decreases below the critical value . Similar to the procedure used in Brauckmann & Eckhardt (2013b), we determine the critical values

(18)

from the intersection of a linear fit to the outer fluctuations with the base level. For and , the re-evaluated onsets conform with the old values within the uncertainties. Note that the difference between the fluctuation amplitudes is less pronounced for than for (cf. figure 4), as expected from the weaker curvature in the first case. Moreover, the enhanced outer fluctuations due to turbulent bursting clearly differ from the situation with stationary outer cylinder or with co-rotating cylinders () where the fluctuation amplitudes are similar for both cylinders.

Figure 5: Onset of enhanced outer fluctuations. The critical rotation ratio for the onset is shown as a function of the radius ratio . Numerical values for are extracted from figure 4 and the value from the TC-experiment (van Gils et al., 2012) was deduced from a bimodal distribution of angular velocities. The line indicates the prediction from equation (17). As the radial partitioning disappears for , we indicate the resulting uncertainty in the onset of enhanced fluctuations by continuing the curve with dashes for .

Figure 5 shows that the prediction from (17) agrees well with the onset of fluctuations in DNS data for (figure 4). Moreover, also agrees with the critical rotation ratio for the occurrence of the bursting, detected as a bimodal distribution of angular velocities in the experiments of van Gils et al. (2012) for .

Figure 6: Enhanced outer fluctuations for as a function of the radius ratio , assessed by the standard deviation of temporal torque fluctuations . (a) Ratio of computed at the outer and at the inner cylinder versus the shifted rotation ratio , where denotes the critical value for the onset of enhanced fluctuations from figure 5. For , was subtracted instead of . (b) Typical fluctuation ratio calculated by interpolation at marked by the vertical line in (a). The grey line represents a best fit to the data and indicates the disappearance of the fluctuation asymmetry near .

In figure 6(a), we extend the analysis of the torque fluctuation amplitudes to the low-curvature radius ratios , and . For the sake of visual clarity when comparing all investigated we show the ratio between the standard deviation at the outer and at the inner cylinder. A value greater than one indicates enhanced outer fluctuations. Moreover, the rotation ratio is shifted by the critical value in order to align the respective regions of enhanced outer fluctuations. The small difference for is the reason why we omitted a similar analysis as in figure 4 for the three highest radius ratios. To quantify the fluctuation asymmetry for strong counter-rotation, we define a typical fluctuation ratio at the shifted rotation ratio for each radius ratio. Figure 6(b) shows that this asymmetry measure, , decreases monotonically with towards a value of which signifies equal fluctuation strengths near the inner and outer cylinder. The fluctuation asymmetry vanishes for a radius ratio of , which does not differ significantly from . This suggests that small differences between inner and outer cylinder turbulence exist for all . Therefore, in this case curvature has an effect for any .

The disappearance of the fluctuation asymmetry may also be related to the local stability properties of the flow. The results in figure 6 suggest that the partitioning into an unstable inner part and a stabilised outer part disappears as . However, the neutral surface for counter-rotation, where the velocity profile passes through zero, discriminates the stability regions and exists for all . Therefore, the radial partitioning has to vanish in a different way for , as we will see from local stability results. Eckhardt & Yao (1995) investigated the evolution of local perturbations to the laminar Couette profile along Langrangian trajectories. They calculated radially dependent eigenvalues

(19)

for the perturbation modes with radial and axial wavenumbers and (see also the local stability results by Dubrulle (1993)). Note that the -dependent factor in the second term of (19) corresponds to Rayleigh’s stability discriminant. In the following, we focus on the largest (most unstable) eigenvalue that can be calculated for given and by maximizing at the most unstable radial position . For that purpose, we minimize the viscous damping (first term in (19)) by selecting the smallest possible radial wavenumber

(20)

Subsequently, we determine the wavenumber ratio that maximizes equation (19). With these optimal values for and , we study the radial variation of the largest eigenvalue .

Figure 7: Radial dependence of the centrifugal instability for and (i.e. ) where the neutral surface is located in the centre of the cylinder gap. The real part of the largest eigenvalue from the local stability analysis along Lagrangian path (Eckhardt & Yao, 1995) is shown for various radius ratios . The radial partitioning of stability decreases with increasing (indicated by the arrow).

To analyse the local stability for strong counter-rotation, figure 7 shows radial profiles of for and the specific rotation ratio . This value is comparable to the value used in the study of the fluctuation asymmetry in figure 6. The positive real part of the eigenvalue inside the neutral surface indicates instability in the inner partition while the outer partition is vicously damped. Most importantly, the radial variation of the local stability decreases with which is reminiscent of the disappearance of the radial differences in the fluctuation behaviour. For large , the eigenvalues move in opposite directions: in the inner partition, they become less unstable and approach neutral stability from above, whereas in the outer partition, they approach the neutral value from below.

The results of this section show that the radial partitioning of the flow that gives rise to enhanced outer fluctuations conforms with the prediction (17) for . However, this curvature effect as well as the radial variation of the local stability disappear with vanishing curvature for . The disappearance of the intermittent bursts in the outer partition is also relevant for an understanding of the variation of torque, since the appearance of these bursts has been linked to the emergence of a torque maximum (van Gils et al., 2012; Brauckmann & Eckhardt, 2013b). The absence of this intermittency should therefore change the variation of torque with rotation, as we now discuss.

4 Variation of the momentum transport

Figure 8: Variation of the momentum transport ( with changing mean rotation () calculated in DNS for the radius ratios specified in the sub-plots. The simulations are performed for ( ), () and ( ). The open symbols in (f) show results from simulations of RPCF. Note the increased -scale in sub-figures (a) and (b).

We now turn to the rotation dependence of the mean torque with increasing and with variations in radius ratio. Figure 8 shows the Nusselt number , the torque in units of its laminar value. For the wide-gap TC system with , the torque maximum initially occurs at for and then shifts to for corresponding to as analysed before (Brauckmann & Eckhardt, 2013b; Merbold et al., 2013). For this shift in the maximum is more pronounced, with varying from at to at (figure 8b). For even higher , the experiments of Paoletti & Lathrop (2011) and van Gils et al. (2011, 2012) show no further shift in the position of the torque maximum so that is likely to be close to the asymptotic value for high . As discussed in §3, these high- torque maxima for and coincide with the onset of intermittent bursts in the outer partition.

The situation changes drastically for low-curvature TCF with as shown in figures 8(d)–(f): instead of a single maximum, one notes a broad torque maximum near and a second narrow maximum near . This narrow maximum emerges with increasing shear and, at , is similar in magnitude to the broad maximum. Indications for the narrow maximum were first seen by Ravelet et al. (2010)2 in a low-curvature TC experiment (): Their data show a very slight bump in the torque near for and and a monotonic increase with for . However, their figure 8 only shows torque measurements for so that the broad maximum found here at lies outside their investigated range. The presence of two maxima, a broad one and a narrow one that emerges with increasing , was also observed in RPCF (Salewski & Eckhardt, 2015). They compare well to the maxima in low-curvature TCF as demonstrated for in figure 8(f).

Figure 9: Comparison of the momentum transport () for various radius ratios as a function of the mean rotation () at . The values are compensated with a scaling in employing the scaling exponent measured by van Gils et al. (2012). Sub-figure (b) details the narrow maximum in (a). For purposes of clarity, the results for are only included in (b) where the ones for and are left out. The open stars labelled by show results from RPCF simulations. The inset in (a) compares our DNS results for to the TC-experiment with and of a few (Ostilla-Mónico et al., 2014a) and reveals a slight shift of the maximum with increasing .

Finally, the torque for (figure 8c) shows a behaviour intermediate between that of the narrow-gap and the wide-gap TC systems. While the broad torque maximum known from systems with larger radius ratios is present, the second maximum around is not as narrow and as clearly visible as for . On the other hand, the extrapolation of equation (17), which captures the maximum for and , to the case predicts a maximum at and . This is close to the low-curvature narrow maximum and suggests that the two will interfere. Indeed, the magnification in figure 9(b) of the region in around the low-curvature narrow peak shows both, a slight maximum at and a broader one around .

In figure 9, we show the torque as a function of the rotation number for various radius ratios. In order to compensate for the slightly different of the compared torques, we use the observation by Dubrulle et al. (2005) that the torque variation can be decomposed into a scaling with modulated by an amplitude that describes the dependence on and we thus normalise the torques by . For , the Nusselt numbers for different radius ratios collapse. The figure also contains data for RPCF which also agree nicely with those of low-curvature TCF. Data collapse is also observed with other quantities. Dubrulle et al. (2005) previously noted that the torque normalized by its value for stationary outer cylinder varies only little with when plotted as function of . Similarly, Paoletti et al. (2012) found a collapse of another torque ratio as a function of for various radius ratios. Our figure 9 shows that up to a scaling with a normalisation by the laminar torque suffices to compensate for the curvature dependence and to achieve the collapse for . Note that corresponds to so that the normalisation by has an additional -dependence compared to the normalisation by . The collapse of the data in all three cases is of similar quality and does not favour one particular normalisation. However, plots with respect to (not shown here) do not give such a collapse, confirming that is the appropriate parameter in which to describe the transition from TCF to RPCF.

For lower , the collapse of the torques is limited to the region for and to for and . For these rotation numbers, turbulent bursting does not occur in the outer partition. For smaller , where the bursting in the outer partition occurs, the Nusselt numbers in figure 9(a) depend on , suggesting that only this radial flow partitioning introduces the strong curvature dependence of the torques. In conclusion, the impact of the global rotation (parametrised by ) on the turbulent momentum transport becomes independent of if the flow is not partly stabilised by counter-rotating the outer cylinder or if this curvature effect becomes negligible for large radius ratios.

The position of the torque maxima for and , which in is given by constant rotation numbers and , translates into -dependent rotation ratios , which are represented by the thick solid lines in figure 10. The torque maximum locations from the DNS (triangles and squares) closely follow these lines for . In particular, this reveals that for both the broad and the narrow torque maximum occur for co-rotating cylinders, i.e. . Therefore, the narrow maximum cannot be related to the detachment mechanism that causes the torque maximum at counter-rotation for and . For the latter radius ratios, the maximum location agrees with the predictive line in figure 10. Finally, the intermediate radius ratio shows indications of all three torque maxima close to the lines in figure 10, with the detachment and the narrow maximum lying close together and being not as pronounced as the broad maximum, cf. figure 9.

Figure 10: Ratio of angular velocities of maximal torque as a function of the curvature number . The open circles, triangles and squares summarise our numerical results for the detachment, broad and narrow maximum, respectively, at from figure 9. The maximum locations were calculated from quadratic fits to the peaks in , except for the narrow maximum for , which is not well approximated by a parabola and therefore determined by the location of the largest data point. The small circles indicate the experimental results by Ostilla-Mónico et al. (2014a) for to . These are complemented by the prediction from equation (17) (Brauckmann & Eckhardt, 2013b) and by the angle bisector by van Gils et al. (2012). In addition, two -lines for constant rotation number and are shown, corresponding to the broad and narrow torque maxima.

In addition, figure 10 shows the torque maxima identified for various radius ratios in experiments at much higher Reynolds numbers (Ostilla-Mónico et al., 2014a). While the maximum positions for , and conform with the trend of the detachment maximum, the one for clearly deviates from both and our simulations; this will be discussed in detail at the end of this section. Moreover, the angle bisector line , that was suggested by van Gils et al. (2012) for the position of the maximum , lies close to the experimental torque maxima. However, clearly deviates from the torque maximum for (Merbold et al., 2013) and differs in its functional behaviour for from our DNS results.

For the range of shear Reynolds numbers investigated here, the rotation dependence of the torque significantly changes with increasing , c.f. figure 8. It is likely that this transformation process continues beyond . However, concerning the detachment torque maximum for and it was shown that, after an initial transformation with increasing shear, the rotation dependence of the simulated torques at already compares well to experimental torque measurements at much higher to (Brauckmann & Eckhardt, 2013a; Merbold et al., 2013). On the other hand, concerning the two torque maxima in low-curvature TCF (), Ostilla-Mónico et al. (2014a) find in their torque measurements for and that the power-law exponent of the torque scaling depends on the system rotation for below a few , which indicates that the transformation of the rotation dependence still continues up to . As regards the second narrow torque maximum, we expect that it grows further in relation to the broad maximum for and that this may shift the torque-maximising rotation number a bit.

The torque maximum from the experiment at (Ostilla-Mónico et al., 2014a) supports this assumption as shown in the inset of figure 9(a) which compares experiment and DNS: The experimental torque-maximising rotation ratio corresponds to which only slightly deviates from for the narrow maximum at lower . In conclusion, apart from small changes with , our simulations already capture the beginning of the general turbulent behaviour at much higher shear rates.

4.1 Torque due to vortical motion

In this and the following sections, we investigate the momentum transport characteristics that underlie the rotation dependence of the Nusselt number discussed above. Vortical motions, such as Taylor vortices and their turbulent remnants, are known to effectively contribute to this transport by moving fast fluid from the inner cylinder outwards and slow outer fluid inwards. Their effects on the turbulence have been discussed previously (Lathrop et al., 1992; Lewis & Swinney, 1999; Martínez-Arias et al., 2014). Brauckmann & Eckhardt (2013b) quantified the link of the vortical motion to the torque and found that the torque contribution of the mean vortical motion was largest near the torque maximum. Flow visualizations of DNS by Salewski & Eckhardt (2015) show distinct vortical states which underlie the narrow and broad maxima for RPCF. Furthermore, using experiments at (Huisman et al., 2014) and DNS at (Ostilla-Mónico et al., 2014b), these structures have been detected over a range of rotation numbers where their presence is associated with a single torque maximum.

To investigate the effects of vortical structures to the torque, we first extract the mean vortical motion underlying the turbulence in the full simulation and measure its contribution to the momentum transport. The mean vortices consist of temporally and streamwise averaged turbulent Taylor vortices and are analogous to the “large-scale circulation” found in turbulent Rayleigh-Bénard convection (Ahlers et al., 2009). Next, since Taylor vortices do not depend on the streamwise direction, we also consider the torque which results from streamwise independent flow. We then summarize our results in figure 11.

Mean vortical motion

We follow the procedure introduced in Brauckmann & Eckhardt (2013b) and decompose the flow of the full DNS into a mean contribution (or for RPCF) that includes the mean vortical motion in the -plane (-plane) and into the turbulent fluctuations around these mean vortices . Substituting this velocity decomposition into the momentum flux equation (12) results in a separation of the torque into a contribution that is caused by the mean vortical motion and a second contribution that is due to turbulent fluctuations , i.e. with

(21)

Analogous expressions for the decomposition of the driving force into can be obtained for RPCF by substituting the velocity decomposition into the momentum flux equation (15). To capture the correct amplitude of the turbulent Taylor vortices, we fix their spanwise position during the temporal average of and in (21) by correcting a potential spanwise drift. Such a drift would lead to a cancellation of the large-scale motion over time, even though it might be strong. Note that while the total transport is constant over , the individual terms in (21) vary with the radius. This motivates the additional radial average in (21) in order to quantify the typical strength of the torque contributions. Since we are interested in the mean vortical motion which dominates in the central region and not in the boundary layers, we restrict this average to the range between and .

In figure 11, the torque due to the large-scale circulation is compared to the total torque . For all cases, exhibits a maximum. For and RPCF, the maximum of , where the momentum transport by the mean vortices is most effective, nearly coincides with the narrow maximum of the total torque . Moreover, it is apparent that a distinct narrow peak also occurs in mean-vortex torque for large . This suggests that the narrow, high- maximum is linked to an efficient large-scale circulation. For lower radius ratios, i.e. and , the coincidence between the maxima in and is not as significant; nevertheless, as the analysis in Brauckmann & Eckhardt (2013b) demonstrates, this coincidence does exist for and when the radial average in (21) covers the complete radial gap instead of the central region. Finally, the mean-vortex torques in figure 11 reveal that the mean Taylor vortices occur and grow in strength for as previously observed in the experiment by Ravelet et al. (2010).

Importance of streamwise invariant structures

We perform simulations that force the flow to be invariant in the downstream direction by taking no Fourier mode in the direction (direction for RPCF). For TCF this corresponds to axisymmetric simulations. Note that this procedure differs from simulations in a cross-section because three velocity components are still active, however, allowing only for spatial variations in two directions (wall-normal and spanwise). These simulations of streamwise invariant vortical flow result in torques, denoted as , that we compare to the total torques in figure 11.

Figure 11: Comparison of the total torque ( ) to the torque that results from the streamwise invariant flow () and to the torque due to the mean flow ( ) for . All torques are measured in units of the laminar torque giving Nusselt numbers . The open symbols in (f) show corresponding results from RPCF simulations.

The torque also shows a broad maximum, which has an -dependent shift in its location: for , the maximum is at ; and for , it is near . For some radius ratios, namely , , this places the -maximum at nearly the same as the broad maximum in the total torque; however, the streamwise invariant simulations overestimate the amplitude of the maximum. In fact, for and RPCF, the -maximum resembles the broad maximum in the total torque. Furthermore, it also appears to agree with the plateau that occurs in the total torque beside the actual torque maximum for (figure 11b). This suggests that the plateau for as well as the broad maximum at lower (cf. figure 8b) follow from the same mechanism that causes the broad maximum for .

The analysis reveals several features of the contribution of vortical motion to the torque for different degrees of curvature. The streamwise-averaged mean flow () appears to reproduce the narrow torque maximum, but this is not reflected by the streamwise-invariant flow (). This seems to be counter-intuitive when considering that is streamwise invariant like the simulations underlying , but neither the detachment maximum in the total torque for nor the narrow maximum for occur in the torques from the streamwise invariant simulations. As a consequence, these maxima must arise from more complicated flows which allow for streamwise fluctuations and cause an additional increase of the momentum transport. The reasons for the strong momentum transport differ between the broad and narrow torque maximum as section §5.3 will illustrate: strong streamwise vortices as well as a high correlation between the radial flow and the angular momentum in these vortices are required for an effective momentum transport and, thus, high torque.

5 Flow characterisation

In section 4 we discussed that the momentum transport converges to a universal behaviour for the low-curvature TCF (), in that the rotation dependence becomes independent of . Furthermore, the dependence on agrees with that observed in RPCF. In the following, we analyse how the convergence for extends to other flow characteristics such as mean profiles and turbulent fluctuations. Moreover, we compare our TC results to the behaviour in RPCF.

5.1 Angular momentum profiles

We analyse here the mean profiles of the specific angular momentum which is radially transported between the cylinders according to the continuity equation (12). To compensate for the varying global system rotation, we show rescaled profiles in figure 12 where and denote the specific angular momentum of the inner and outer cylinder. As a result, the profiles for various radius ratios that belong to the same rotation number collapse as long as the flow is not subject to the radial partitioning of stability for moderate counter-rotation (cf. section 3): The rotation number corresponds to co-rotation () for all investigated , and all profiles of in figure 12(d) collapse well. However at , only the simulation for with shows the bursting in the outer partition and a deviating profile in figure 12(c). Similarly at (figure 12b), the profiles for corresponding to -values in the bursting regime clearly deviate from the profiles for and with which are closer together. Finally, no collapse is observed for which always corresponds to exact counter-rotation. In addition, figure 12 shows profiles from RPCF with the system rotation subtracted, i.e. , that agree well with the collapsed profiles from TCF when rescaled to the same interval. The reason for this correspondence between angular momentum profiles in TCF and in RPCF will be discussed in the next section.

Figure 12: Profiles of the rescaled specific angular momentum for the mean system rotations defined by and for six different radius ratios. The corresponding RPCF profile