Transition to the ultimate turbulent regime in a very wide gap Taylor-Couette flow (η = 0.1) with a stationary outer cylinder

The boundary layers in turbulent Taylor-Couette flow are exposed to transitions from laminar to turbulent states if the flow is sufficiently sheared. The present study examines this particular transition from the so-called “classical” to “ultimate” regime experimentally for a very wide-gap Taylor-Couette flow with a radius ratio of and shear Reynolds numbers of up to . In order to determine the transition, the angular momentum transport is measured by using torque sensors at the inner wall. This is complemented by measuring the radial and azimuthal velocities via a time-resolved Particle-Image-Velocimetry (PIV). The transition to the ultimate regime is found at . The dimensionless angular momentum flux showed an effective scaling of for and is in agreement with the scaling laws used for the ultimate regime in narrow-gap Taylor-Couette flows. In addition, a spectral analysis was performed showing the existence of highly energetic small-scale and large-scale patterns in the classical regime whereas only highly energetic large-scale patterns were observed in the ultimate regime.

Introduction. -The flow between two coaxial independently rotating cylinders has been used for several decades as a model to study shear flow instabilities, transitions to turbulence, flow patterns, and many more phenomena, see [1][2][3]. This particular system is referred to as Taylor-Couette (TC) flow and is characterized by the radius ratio η = r 1 /r 2 and its aspect ratio Γ = L/d, where r 1 and r 2 are the inner and outer radii of the respective concentric cylinders, d the enclosed gap width between both cylinders defined by r 2 −r 1 and L the length of the system. The flow confined between the cylinders is driven by the independent rotation of each cylinder, resulting in parameters describing and quantifying a shear driving flow by the rotation ratio μ = ω 2 /ω 1 , where ω 1,2 is the rotation velocity of the inner and outer cylinder, denoted by the subscript 1 and 2, respectively. This predefined condition gives rise to a well-known non-dimensional parameter, the (a) E-mail: hamede@b-tu.de (corresponding author) shear Reynolds number written as [4] Re s = 2 1 + η |ηRe 2 − Re 1 | = 2r 1 r 2 d (r 1 + r 2 )ν |ω 2 − ω 1 |, (1) where ν is the kinematic viscosity of the fluid. In the case of cylinders rotating differentially, the global transport quantities between the cylinders gain the attention of the researchers, as such systems and behaviours are found in different engineering applications (e.g., journal bearings, rotating turbines, etc.) and also astrophysical ones (e.g., accretion disks). Eckardt, Grossmann, and Lohse (EGL) [5] showed that the conserved transport quantity in the TC flow is the angular momentum flux between the inner and outer cylinders, and it is defined by where u r is the radial velocity component, u φ azimuthal velocity component, ω = u φ /r the angular velocity and A(r),t denotes as the mean over a cylindrical surface at radius r and over time, t.

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To study the transport of angular momentum, one has to consider two main aspects in TC flow. The first is the influence of rotation ratio with fixed Re s . In this case the angular momentum achieves a maximum at a specific rotation ratio, see [6][7][8][9][10]. The second aspect is the study of the scaling of the angular momentum transport by the flow shear rates, at fixed rotation ratios. For this purpose different analyses have been carried out to study the similarities between TC flow and Rayleigh-Bnard (RB) convection. RB convection refers to a bottom heated and top cooled fluid layer that is driven by buoyancy, see [5,[11][12][13]. By comparing both TC and RB the the angular momentum transport of TC flow could be referred to as a quasi-Nusselt number N u ω = J ω /J ω lam and was thus the analogy to the heat transport in RB flow, where J ω lam = 2νr 2 1 r 2 2 (ω 1 − ω 2 )/(r 2 2 − r 2 1 ) is the transported flux in laminar flow conditions. Since then the N u ω can be directly linked to the TC system's torque, T , given by the inner cylinder in order to keep its rotation velocity, ω 1 , constant. A dimensionless torque can be defined by G = T /2πlρν 2 , where ρ is the fluid density, to define the quasi-Nusselt number with N u ω = G/G lam , where G lam = r 1 r 2 Re s d −2 is the dimensionless torque for the laminar case. While both TC flow and RB convection can reach an "ultimate" state, a second similarity is seen where turbulence is fully developed in the bulk and within the boundary layers [6,14]. The transition to the ultimate regime occurs when the boundary layers are sheared strong enough that they undergo shear instability and hence become turbulent [6,14]. The transition to the ultimate turbulent regime can be characterized by an effective scaling between the shear Reynolds number and the quasi-Nusselt number where a scaling of N u ω ∼ Re 0.78±0.06 s was investigated for flows in the ultimate regime for η = 0.716 [6,15]. The scaling appears to be approximately the same for values of η = 0.5 with 0.75 ± 0.03 by Merbold et al. [7] and for the same radius ratio by Froitzheim et al. [8] who measured a scaling exponent of 0.76 ± 0.08 by using a different measurement technique. These results suggest that the scaling exponents of the ultimate regime may be independent of η. When considering wider gaps, Burin et al. [16] calculated an angular momentum transport of η = 0.35 using Laser Doppler Velocimetry (LDV) and found a scaling exponent of 0.6 ± 0.1 for 2 × 10 3 ≤ Re 1 ≤ 10 4 and 0.77 ± 0.07 for 2 × 10 4 ≤ Re 1 ≤ 2 × 10 5 . Considering wider gap geometries with η less than 0.35, the existence of such scaling and the transition to the ultimate regime have not yet been observed.
In this letter, we investigate the dependence of the flow on the shear Reynolds number in a very wide gap of η = 0.1. The study considers flow states of up to Re s = 1.5 × 10 5 in purely inner cylinder rotation regime, where the outer cylinder is kept at rest. The radial and azimuthal velocities of the flow are measured at different heights by high-speed particle image velocimetry (HS-PIV). In addition, the torque is measured for very small shear Reynolds numbers. One of the goals of this study was to find the critical shear Reynolds number (Re s,c ) where the flow is transformed to the ultimate regime. This is done by calculating the effective scaling between the shear Reynolds number and the Nusselt number, assuming a scaling exponent of 0.76 ± 0.08 to be the indicator of the transition. The flow properties are studied and compared for the cases before and after the transition, and the angular momentum transport is quantified.
Experimental setup and measurement procedure. -The TC apparatus in this study is the same as the one used by Merbold et al. [17] and consists of an inner cylinder with radius r 1 = 0.007 m and an outer cylinder with radius r 2 = 0.07 m, resulting in a radius ratio of η = 0.1, and gap width of d = 0.07 − 0.007 = 0.063 m. The apparatus length is L = 0.7 m giving an aspect ratio of Γ = L/d = 11.11. In this study, two different measurement techniques are used to evaluate the angular momentum transport. The first technique uses a shaft-to-shaft torque sensor in order to measure the torque applied to the inner cylinder. This enables the calculation of the angular momentum transport. For this purpose, a Lorenz DR-3000 shaft-to-shaft torque sensor is used, which has a measurement range of up to ±0.1 Nm and an accuracy of 0.05%. Unfortunately, the torque sensor could only be used for a few experiments up to a shear Reynolds number of 7000 as the accuracy of the system did not give reliable values until it entirely broke. As it was not possible to rectify this system, further direct torque measurements could not be obtained. In order to calculate the angular momentum transport in larger Re s flows, the radial and azimuthal velocities are measured using HS-PIV at different levels of height. The measurements are conducted in the axial range between ±40 mm starting from the apparatus mid-height, with an axial distance between the two consecutive measured planes of Δz = 10 mm for experiments up to Re s ≤ 3.7 × 10 4 , and Δz = 5 mm for flows with higher Re s . Changing the axial distance between the measured planes was motivated by the visualization performed by Merbold et al. [17], which shows less flow axial dependence for Re s ≤ 3.7 × 10 4 than for higher Re s . At each height, the flow field was recorded for 10 seconds with an acquisition frequency of 200 frames per second (fps) providing 2000 PIV images. The authors refer to [18] for additional information about the measurement and calibration procedures. The angular momentum transport was calculated from the measured velocities using eq. (2), and given in terms of N u ω . By definition, the angular momentum flux J ω is a conserved quantity over the different gap radial positions, as shown by [19], so the value of J ω is constant at the different radial positions through the gap. In this work, the J ω radial profile shows mostly a flat behaviour in the bulk forr = (r − r 1 )( This was a resulting fact of the limitations of the PIV technique to temporarily and spatially scales, especially close to the wall-bounded shear flow. Furthermore, taking into consideration the minor axial dependence of the observed flow [17], and for more detailed temporal analysis, additional measurement sets are conducted at one single height (the mid-height of the apparatus) for the same rotation ratio μ = 0 and different Re s for 40 seconds, resulting in 8000 PIV images. Hence, it is considered that most of the different flow structures with different scales pass through this axial position and are covered by the velocity measurements. These measurement sets are used to perform a spectral analysis of the energy distribution over the different flow scales.
Results. -In this study, the angular momentum transport is studied for different Re s flows with pure inner cylinder rotation. Figure 1(a) shows the variation of N u ω with respect to Re s , the uncertainty bars in the figure show the maximum and minimum values of N u ω in the radial domain 0.3 <r < 0.7, where N u ω (r) is radially averaged. In addition, an increase in the angular momentum transport is noted in terms of in N u ω , with shear Reynolds number Re s . Figure 1 The experimental data suggest that for Re s,cr = 2.5 × 10 4 a transition is achieved in the scaling exponent to 0.76, which is a sign of the flow transition from the classical turbulent regime (where the flow in the bulk is turbulent but laminar in the boundary layers) to the ultimate turbulent regime (both the bulk and boundary layers are turbulent) [7,15,20].
In order to study how the increase of the shear Reynolds number influences the general flow behaviour, in particular when the flow achieves a transition from the classical to the ultimate regimes, the flow velocity profiles are studied. Figure 2 shows the radial profiles of spatially and temporally averaged (t, φ, z), angular velocity (ω = u φ /r) and angular momentum (L = u φ r). The profiles in the figure are presented in their normalized form, defined asω = ω/ω 1 andL = L/L 1 . Figure 2(a) shows the profiles of the angular velocities which show a weak dependence on the shear Reynolds number and tend to mostly collapse to one point. Although a small deviation from this trend is observed in the arear ∈ [0.15, 0.4] for flows with Re s < Re s,cr that are defined by the classical turbulent regime. This deviation decreases as Re s approaches Re s,cr . In contrast to the angular velocity, the angular momentum is more likely to show a distinct dependence on Re s . Figure 2(b) shows the radial profile of L , where the profiles belong to two different regimes with different behaviour. For Re s ≥ Re s,cr the profiles are mostly flat through the gap, while for Re s < Re s,cr the profile shows large variations in the boundaries with a flat behaviour in the middle. Here, this variation in the boundaries decreases as Re s approaches Re s,cr . The deviation from the flat profile, especially near the cylinder walls, is more pronounced near the outer cylinder, as shown in fig. 2(b). It is expected that these deviations are due to bursting or plume injection. The origin of this deviation is further discussed below using spectral analysis. In addition, for Re s > Re s,cr the value ofL ≈ 0.5 was observed in the bulk of the gap. This value was also found by Froitzheim et al. [9] for η = 0.357 and Re s ≥ Re s,cr , where the transition to the ultimate regime was observed, which is in agreement with our finding. TheL ≈ 0.5 value corresponds to (L 1 + L 2 )/2, which indicates that the angular momentum is well-mixed [21]. To understand the secondary flow and its contribution to the angular momentum transport, it is important to study the behaviour of the wall-normal velocity (u r ) fluctuations, the component in which the angular momentum is advected. By definition, the spatial-temporal average of this component is zero, so the spatial-temporal fluctuations of the radial inflow and outflow are of interest. The intensity of these fluctuations is quantified by the root-mean-square of the radial volume flux per height 2πru with u the velocity fluctuation component, u (r, φ, t) = u(r, φ, t)− u(r, φ, t) t . Figure 3 depicts the radial profile of the axially averaged azimuthal and temporal root-mean-square of the radial velocity fluctuations defined as U r rms = σ t,φ (u r (φ, r, t, z)) z for different Re s . The presented profiles are normalized by the shear velocity u s = (2/(1 + η))|u φ,2 − u φ,1 | in order to compensate for the shear driven. As a radial velocity decreases radially due to increasing circumference and incompressibility, the local results are given in volume flux per cylinder height so thatŨ r rms profiles are multiplied by 2πr. Similar to the angular momentum profiles, the profiles show a change in behaviour at the transition to the ultimate regime Re s > Re s,cr . For Re s ≤ Re s,cr the radial fluctuations are strong at radial positions close to the inner cylinder and decrease outwards. The highest relative in-and outflow can be observed for the lowest presented Re s . For the ultimate regime, the profiles are mostly flat across the bulk with small variations towards the boundaries, with the weakest relative in-and outflow fluctuations for the highest Re s observed.
Another flow feature that distinguishes the ultimate from the classical turbulent regime is the scaling behaviour of the various turbulent structures in the flow. In this study, the spatial and temporal energy co-spectra are used to analyse the length scales of different flow structures. The energy co-spectra are studied for two velocity fluctuations u r and u φ , at different fixed radial positions (r = 0.35, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9). The spatial co-spectra are studied at the different azimuthal points n φ = {0, 1, 2, . . . , N − 1}, spaced by the arc length interval Δs = Δφr. The temporal co-spectra are studied at different temporal positions, spaced by the sampling frequency of 200/sec.
The discrete spatial FFT (U r (n φ ) and U φ (n φ )) of both velocity components fluctuations (u r and u φ ) are calculated to evaluate the azimuthal energy co-spectra E rφ by using the algorithm presented in Press et al. [22]. For the spatial co-spectrum, the wave number vector is defined by k n φ φ = (Δs) −1 n φ /N . The spatial energy co-spectrum is calculated for each time step and then averaged over the 8000 snapshots for each presented case. In order to enable the comparison of the co-spectra between the different Re s cases, the co-spectra are normalized based on the trapezoidal integration method by the area A E below its graph Figure 4 shows the co-spectral energy profiles multiplied by the wave number vector k φ and the pre-multiplied temporally averaged azimuthal energy co-spectra shown for different Re s at different radial positions. First, it needs to be said that the measurements cover one third of the gap's azimuthal length, resulting in an arc size at radial positionsr ≤ 0.35 smaller than the gap width. Hence, the results are given for radial positionsr ≥ 0.35, to prevent under-resolved spectral profiles.
For the classical turbulent regime, fig. 4(a) shows the pre-multiplied energy co-spectra for Re s = 10 4 at different radial positions. The different curves belonging to different radial positions are showing majorly the same behaviour with some variations across the gap. A clear peak is observed at a large wavelength k φ d ≈ 15.5 and refers to the amplification of small-scale structures that dominate the flow. The amplitude of these peaks decreases closer to the outer cylinder. These small-scale patterns can be described by structures that exhibit from the outer cylinder boundary layer. For larger wave numbers the spectral energy decays strongly for all radial positions. A second peak is observed for k φ d ≈ 1.2 indicating patterns of large-scale circulation that tend to approximately fill the whole gap. For Re s = 6 × 10 4 flows in the ultimate turbulent regime the pre-multiplied energy co-spectra profiles show different behaviour. No further peaks at large wave numbers are observed; either they no longer exist or they are shifted toward large wave numbers above the measured range. The large-scale peak still exists, but is shifted to k φ d ≈ 2.2 for all radial positions exceptr = 0.9 where no clear peak is observed. Thus, the energy is distributed more homogeneously compared to the classical turbulent regime case and the co-spectra show a negative slope for k φ d ≥ 7 scaled by y = x −5/3 at almost all radial positions. By remaining in the ultimate turbulent regime, fig. 4(c) shows the pre-multiplied energy cospectra for Re s = 1.5 × 10 5 where no peaks are observed for small scales. However, one broad peak is observed for larger scales for all radial positions and is increasing more broadly with the radial position approaching the outer cylinder. The peak can indicate different structures with different large scales in the flow that are all shorter than the gap width. It is clear from the presented profiles, and in contrast to Re s = 6 × 10 4 , that the energy decay rate is dependent on the radial position. For the radial positions closer to the outer cylinder, a stronger energy decay is observed.
The pre-multiplied energy co-spectra for the different investigated flows with different Re s is presented in fig. 4(d),(e) at three radial positionsr = 0.4, 0.6, and 0.9. At these selected positions, the flow was classified into three groups according to the co-spectral shape. The first is the classical regime, where the energy achieves two peaks, one related to small-scale and the other to large-scale structures. The amplitude of the first peak related to the small structures decreases as Re s approaches Re s,cr . The second group is the ultimate regime with 3 × 10 4 ≤ Re s < 8 × 10 4 , and the third is the ultimate regime with Re s ≥ 8 × 10 4 . The main distinction between the second and third groups is that the third has a wider peak for low wave numbers indicating a larger developed turbulence with high energetic patterns and a broad range of large sizes. Furthermore, the energy decaying rate at larger wave numbers shows a radial dependence for the third group, but not for the second.
To understand the temporal behaviour of the flow for the different studied parameters, the temporal co-spectra profiles are presented in fig. 5 for flows with different Re s at different radial positions. The azimuthal energy co-spectrum is calculated at each azimuthal step Δφ and averaged over the different azimuthal points. Furthermore, in order to reduce the noise that is related to the inner cylinder's rotation without altering the profile shape, the profiles in fig. 5 are smoothed using a sliding median method. The presented profiles consist of 4000 points, and the smoothing was done in intervals of 70 points. Using the same group classification presented above, fig. 5(a) shows the flow in the classical regime which achieves a first peak at a small frequency ≈ 0.1 Hz, with a frequency peak being independent of the radial position, although the energy decreases as we approach the outer cylinder. The energy decreases for larger frequencies and reaches a broad minimum where it tends to increase again for the highest frequency. It can be assumed from the figure that the first peak with low frequencies is related to large-scale patterns (k φ d ≈ 1.2) that fill the whole gap, as the frequency of the peak is independent of the radial position. The second peak seems to be dependent on the radial positions and is assumed to be related to the smallscale structures (k φ d ≈ 15.5). The second group shows the flow is in the ultimate regime but with low Re s . The profiles in fig. 5(b) show a relatively broad peak for frequencies between 1 and 7 Hz. This can be explained by the variety of patterns with different scales explained above. For the third group in fig. 5(c) mostly the same behaviour is observed, but the broad peak in frequency is extended to higher frequencies up to 10 Hz. The broad spatial and temporal spectral profiles peak in this group are result of the development of the turbulence, so different patterns with a broad variety of sizes and frequencies appear in the flow.
Conclusion. -In this study, the angular momentum transport in a very wide gap η = 0.1 Taylor-Couette flow was quantified for a set of shear Reynolds number. A scaling relation is found between the Nusslet number and the shear Reynold number and is defined by N u ω ∼ Re 0.76 s for Re s ≥ 2.5 × 10 4 . This effective scaling is a measure of the transition from the classical to the ultimate turbulent regime. In addition, the angular velocity is measured and shows mostly the same behaviour in both classical and turbulent regimes. In contrast, the angular momentum profiles show a clear dependence on Re s . For the ultimate flow regime the profiles show flat behaviour in the bulk, while for the classical flow regime, they show high variation in the regions next to the inner and outer cylinders. The latest behavour of the profiles in the classical regime are returned to the injection of small structures into both cylinders. Moreover, the spectral analysis of the flow in the classical regime reveals the presence of highly energetic small-scale structures with large frequencies, and large-scale structures having the size of the gap width with small frequency. However, only highly energetic large-scale structures with a broad variety of moderate frequencies are observed in the ultimate regime, having smaller sizes than the large-scale structures investigated in the classical regime. * * *