Calculation of the mean velocity profile for strongly turbulent Taylor--Couette flow and arbitrary radius ratios

Taylor--Couette (TC) flow is the shear-driven flow between two coaxial independently rotating cylinders. In recent years, high-fidelity simulations and experiments revealed the shape of the streamwise and angular velocity profiles up to very high Reynolds numbers. However, due to curvature effects, so far no theory has been able to correctly describe the turbulent streamwise velocity profile for given radius ratio, as the classical Prandtl--von K\'arm\'an logarithmic law for turbulent boundary layers over a flat surface at most fits in a limited spatial region. Here we address this deficiency by applying the idea of a Monin--Obukhov curvature length to turbulent TC flow. This length separates the flow regions where the production of turbulent kinetic energy is governed by pure shear from that where it acts in combination with the curvature of the streamlines. We demonstrate that for all Reynolds numbers and radius ratios, the mean streamwise and angular velocity profiles collapse according to this separation. We then derive the functional form of the velocity profile. Finally, we match the newly derived angular velocity profile with the constant angular momentum profile at the height of the boundary layer, to obtain the dependence of the torque on the Reynolds number, or, in other words, of the generalized Nusselt number (i.e., the dimensionless angular velocity transport) on the Taylor number.


Introduction
Most flows in nature and engineering are bounded by solid walls. In general, the flow in the immediate vicinity -at a molecular scale distance -from the wall has the velocity of the wall, the so-called no-slip boundary condition. As a consequence, a steep gradient in the mean streamwise velocity profiles exists within the boundary layer (BL) region between the wall and the freely flowing fluid above. In the BL, the action of viscosity against the gradient of the streamwise velocity results in viscous dissipation, the conversion of kinetic energy into heat.

Turbulent flow over a flat plate: Prandtl-von Kárman BL theory
For slowly flowing fluids (low Reynolds numbers), the edge of the BL remains smooth, and the fluid flow in the BL is two-dimensional. This laminar BL is described by the famous Prandtl-Blasius self-similar solution (Schlichting 1979). However, for fast flowing fluids (high Reynolds numbers), the BL becomes turbulent, and the flow inside the BL becomes vortical and three-dimensional. Although exact solutions of these turbulent BLs do not exist, a well-established functional form of the mean streamwise velocity can be obtained based on simple dimensional arguments (Schlichting 1979). The hallmark result therefrom can be obtained from realizing that the mean streamwise velocity gradient in the wall-normal direction ( du dy ) is a function of two dimensionless parameters only (Pope 2000), where u τ is the friction velocity defined as u τ = τ w /ρ, τ w is the mean wall shear stress, ρ is the fluid density, δ is the outer length scale (e.g. the BL thickness), and δ ν is the viscous length scale δ ν = ν/u τ , with ν the kinematic viscosity of the fluid. Nondimensionalization by the viscous scales u τ and δ ν is indicated by a superscript '+'. We define the friction Reynolds number based on these viscous quantities as Re τ = uτ,id 2ν , where d is the gap width between the two rotating cylinders. The yet undefined function Φ( y δν , y δ ) must go to a constant (= κ −1 ) when δ ν y δ, which is known as the inertial sublayer. In this limit, we can integrate (1.1) and arrive at the celebrated logarithmic law of the wall for turbulent BLs over a flat surface: (

1.2)
This law is connected with the names of Prandtl and von Kármán. It is supported by overwhelming experimental and numerical evidence (e.g. Smits et al. (2011)). The values of the two parameters are κ ≈ 0.4 and B ≈ 5.0. An important extension of the theory concerns buoyancy stratified BLs, where an additional forcing acts on the wall-normal momentum component. A prominent example of such a system is the atmospheric surface layer, where thermal forcing stabilizes or destabilizes the flow. The thermal stratification introduces, aside from δ ν and δ, a third relevant length scale: the Obukhov length L ob (Obukhov 1971). This length L ob is proportional to the distance from the wall above which the production of turbulence is significantly affected by buoyancy, and below which the production of turbulence is governed purely by shear. With the introduction of this length L ob , (1.1) becomes: which was first proposed by (Monin & Obukhov 1954). For the inertial sublayer (δ ν y δ) only the dependence on y L ob remains. Various empirical fits exist for Φ( y L ob ). Evidently, in the limit of y L ob 1 they must obey Φ( y L ob ) = κ −1 , thus indicating that buoyancy plays no role. We point to §4 of Monin & Yaglom (1975) for an in-depth analysis of stratified BLs.
1.2. Turbulent flow with streamwise curvature: Taylor-Couette turbulence Whereas flat plate BLs are often studied, and the existence of a logarithmic profile of the mean streamwise velocity is well established, the study of flows with streamwise curvature is less developed, despite its ubiquity, e.g. ship hulls or turbomachinery. In this paper, we attempt to narrow this gap. One canonical system for flow in a curved geometry is Taylor-Couette (TC) flow. TC flow is the shear-driven flow in between two coaxial, independently rotating cylinders. Since the physical system is closed, one can derive a global balance between the differential rotation of the cylinders and the total energy dissipation in the flow, which is directly related to the torque (T ) on any of the cylinders (Grossmann et al. 2016).
The dimensionless torque G is defined as G ≡ T /(ρν 2 L z ), where L z is the height of the cylinder. It depends on the Reynolds number of the inner and outer cylinder, defined as Re i,o = ω i,o r i,o d/ν. Here, r i,o is the radius of the inner (outer) cylinder, ω i,o is the angular velocity of the inner (outer) cylinder, d is the gap width, and ν is the kinematic viscosity. The relation G(Re i , Re o ) is directly connected to the structure of the mean velocity profile. Uncovering this relation -for its fundamental implications and practical relevance -can be considered the primary research question.
In this paper we consider pure inner cylinder rotation (Re o = 0), for which in the laminar case Taylor (1923) derived that G ∝ Re. For intermediate Re, Marcus (1984) -in analogy to the work of Malkus & Veronis (1958) on Rayleigh-Bénard (RB) flow -argued by exploring marginal stability arguments that G ∝ Re 5/3 . He modelled the flow domain as being partitioned into a turbulent bulk region with constant angular momentum L (Townsend 1956) and two laminar BLs. For high but finite Re, the BLs become turbulent (Grossmann & Lohse 2012;Ostilla-Mónico et al. 2015;Krug et al. 2017), and the effective scaling exponent increases with increasing Re (Lathrop et al. 1992a,b). Analogous to the interpretation of the strongly turbulent regime by Kraichnan (1962) and Chavanne et al. (1997) in RB flow, Grossmann & Lohse (2011) derived logarithmic corrections to the G(Re) scaling, coming from the turbulent BLs, such that G ∝ Re 2 × log(Re)-corrections. In the limit of Re → ∞, the dissipation will anywhere in the flow scale with the velocity difference cubed, irrespective of the length scale (Lathrop et al. 1992a), resulting in a torque scaling of G ∝ Re 2 .
High-fidelity data on the structure of the BL are essential for testing all proposed scaling relationships. Therefore, much work has been carried out to determine the mean streamwise velocity profile at high Re. Huisman et al. (2013) used particle image velocimetry (PIV) and laser doppler velocimetry (LDA) to study the turbulent BL at an unprecedented resolution. For η = 0.716, where η is the radius ratio, they find that for high Re i , i.e. Re i = O(10 6 ), the classical logarithmic BL exists only in a very limited spatial region of 50 < y + < 600. van der Veen et al. (2016) employed PIV to study the velocity profiles at low radius ratio of η = 0.50, for which the curvature effects are stronger, and find no von Kármán type logarithmic BL. For η = 0.91, Ostilla-Mónico et al. (2014a) and Ostilla-Mónico et al. (2015) employed direct numerical simulations (DNSs) and find that the slope of the mean streamwise velocity profile is ever changing with Re i , at least up to Re i = O(10 5 ). We further note that Grossmann et al. (2014) argue that the appropriate velocity that obeys the classical von Kármán profile is the angular velocity, rather than the streamwise velocity, based on conservation laws of the Navier-Stokes equations in this axial symmetry.
In this paper we will explain that the introduction of a curvature length scale delineates the region where one can expect a shear-dominated turbulent BL and another region where curvature effects will alter the structure of the flow, similar as the Obukhov length in stratified shear flow separates the shear dominated regime from the buoyancy dominated regime. This paper is organized as follows: In §2 we will give the Navier-Stokes equations and boundary conditions for TC flow. In §3 we will discuss the used datasets. We will then, in §4, derive a functional form for the angular velocity throughout the entire BL for arbitrary Reynolds numbers but only for pure inner cylinder (IC) rotation. We extend the theory towards varying radius ratios in §5. Finally, we match the BL and bulk velocity profiles and arrive at a new functional form for Nu(Ta) and C f (Re i ) for TC in §6. The paper ends with conclusions and an outlook.

Navier-Stokes equations for Taylor-Couette flow
When the inner cylinder rotates and the outer cylinder (OC) remains stationary (the case to which we restrict us in this paper), TC flow is linearly unstable (Lord Rayleigh 1916). The ratio between the destabilizing centrifugal force and the stabilizing viscous force is expressed by the Taylor number (Taylor 1923), The Reynolds number Re i,o is related to Ta via the relation Re Eckhardt et al. (2007) showed that the mean angular velocity flux is independent of r, where . A(r),t refers to averaging over a cylindrical surface A(r) and time t. The torque T per unit length is related to J ω by T = 2πρJ ω . Therefore also T is constant with r. TC flow, see the schematic in figure 1, is described by the three components of the Navier-Stokes equations in an inertial frame in cylindrical coordinates as (Landau & Lifshitz 1987), with w r the radial velocity, u θ the azimuthal velocity and v z the axial where the operators are, with for IC rotation only, the boundary conditions w r (r Note that P t is the kinematic pressure, and ρP t is the physical pressure. The continuity equation reads

Employed datasets
In this paper we apply our theoretical analysis to published datasets with varying radius ratio, see Table 1 in the appendix. We now briefly describe the techniques that are used to acquire these datasets. However, we refer to the original papers for more details. Huisman et al. (2013) did experiments on highly turbulent inner cylinder rotating TC flow with the Twente turbulent TC facility (T 3 C) ( van Gils et al. 2011a), with the radius ratio η = 0.716 and the aspect ratio Γ = 11.7. In particular, they carried out PIV and particle tracking velocimetry (PTV) to measure the mean and the variance of the streamwise velocity profiles at 9.9 × 10 8 Ta 6.2 × 10 12 , for both the IC BL and the OC BL. van der Veen et al. (2016) performed experiments on turbulent TC flow in the classical turbulent regime (i.e., before the BLs become turbulent) with the Cottbus TC facility (Merbold et al. 2013), with radius ratio η = 0.50 and aspect ratio Γ = 20. They carried out PIV to measure the mean streamwise and wall-normal velocity profiles at 5.8 × 10 7 Ta 6.2 × 10 9 . Although van der Veen et al. (2016) carried out both counter rotation and pure inner cylinder rotation experiments, we will discuss here the latter dataset only. Ostilla-Mónico et al. (2015) carried out DNSs of highly turbulent IC rotating TC flow by using a second-order finite-difference scheme (Verzicco & Orlandi 1996;van der Poel et al. 2015). With a radius ratio of η = 0.909 they simulated three cases with 1.1 × 10 10 Ta 1.0 × 10 11 . Additionally, they simulated a large gap case, η = 0.5, with Ta = 1.1 × 10 11 . For all cases the aspect ratio was fixed at Γ = 2π/3.

Velocity profiles in Taylor-Couette turbulence
Whereas effects of spanwise curvature on the profiles were investigated before (Grossmann & Lohse 2017), in this section we set out to develop a new functional form of the mean angular velocity profile ω + (y + ) (with ω + = ω/ω τ and ω τ, Figure 2: A schematic representation of the analogy between the effects of buoyancy and streamline curvature on a BL. (a) A flat plate unstably stratified BL. The volume element, with volume V , top and bottom surface area A, and height ∆z, exemplifies the working of the shear force F s and the buoyancy force F b . Note that β is the thermal expansion coefficient and g is the gravitational acceleration that is defined positive in the −z direction. (b) A top view of a BL over a curved surface (e.g. the TC IC). In analogy to F b in (a), the centrifugal force F c works in the wall normal direction, and in the case of IC rotation, destabilizes the flow. that part of the IC BL and OC BL where the streamwise curvature effects are significant. Note that (1.1) can also be postulated for ω(y), so that the gradient becomes where Φ ω y δν , y δ goes to a constant in the inertial region δ ν y δ. We follow the conclusion of Grossmann et al. (2014), namely that near the wall the angular velocity ω + (y + ) fits to a logarithmic form closer than the azimuthal velocity u + (y + ), and we apply our analysis to ω + (y + ). For reference we have added figure 11 in the appendix, where we apply the analysis (see following pages) to the azimuthal velocity profile. A slightly less convincing collapse of the azimuthal velocity profiles, in comparison to the angular velocity profiles, indicates that the angular velocity profile is indeed the appropriate quantity.
In §4.1 we first derive the curvature Obukhov length and then apply our analysis to the highest Re dataset available (Huisman et al. 2013). Subsequently, we analyse both the IC BL ( §4.2) and OC BL ( §4.4) and in §4.3 also the constant angular momentum region in the bulk.

Derivation of the curvature Obukhov length L c
Following Bradshaw (1969), we draw the analogy between the effects of buoyancy and streamline curvature on turbulent shear flow. Therefore it is informative to assess the balance of turbulent kinetic energy (TKE) in the flow. To do so, we first Reynolds-decompose the velocity and pressure field ((2.3) to (2.5)), such that v = U + u, where v = (w r , u θ , v z ) is the full velocity, U = (W, U, V ) is the time averaged velocity and u = (w, u, v) is the fluctuating component. Upon multiplying the decomposed Navier-Stokes equations by u, and then taking the time average, we arrive at the TKE equations. In vector notation, with the definition of TKE (strictly speaking the turbulent intensity since we divide by ρ) being q = 1 2 (u 2 + v 2 + w 2 ), the TKE equation reads (see also Moser et al. (1984)): (4.2) We consider a statistically stationary flow that is homogeneous in the wall-parallel directions. Further, we assume that the net radial transport of TKE over the boundaries of a volume element in the turbulent BL is zero for δ ν y δ. We then arrive at a reduced form of (4.2), where the net local production of TKE is equal to the local dissipation. 3) The first term on the left-hand-side of equation (4.3) represents the production of TKE due to a gradient of the mean streamwise velocity profile, i.e. shear. The curvilinear coordinate system gives rise to an additional production term (the second term), as compared to turbulent shear flow over a flat boundary. In fact, such additional production terms due to curvature appear both in the u θ -component equation and in the w rcomponent equation, and are respectively, 1 r uwU and − 2 r uwU . Together, they sum up to the second term on the left-hand-side in (4.3).
The process of additional production of TKE by curvature of the streamlines may be explained by the conservation of angular momentum L = U r (Lord Rayleigh 1916;Townsend 1956). If one considers a vortex that exchanges two fluid elements from r 1 to r 2 where r 1 < r 2 , the change in kinetic energy whilst conserving L is ∆E k = 1 2 (U 2 1 r 2 1 − U 2 2 r 2 2 )( 1 ). For (r 2 − r 1 )/r 1 1, the change in E k can be rewritten as where δr ≈ r 2 − r 1 and r ≈ r 1 ≈ r 2 . This is a very similar energy exchange as for buoyancy stratified flows, where δE = βg dT 2 dz 2 (δz) 2 (Townsend 1976). In fact, we see that if dL/dr < 0, the work carried out by the vortex is positive and the IC rotating and stationary OC TC flow might be called unstably stratified (Lord Rayleigh 1916;Esser & Grossmann 1996), whereas for dL/dr > 0 (OC rotating, IC stationary) the vortex requires energy to survive and the flow is stably stratified.
In pursuing this analogy, which we illustrate in figure 2, we expect a region in the flow where (∂ r U U/r) from (4.3) such that the production of TKE is governed solely by shear, and the flow there behaves identical to flat plate BLs. Next to this, another region might exist where the production of TKE is governed solely by curvature effects (U/r ∂ r U ) and curvature stratification effects dominate. The demarcation line that separates the two regions is the location where both mechanisms are of comparable magnitude. Bradshaw (1969) recognized the similarity between buoyancy effects and streamline curvature, and derived the curvature analogy of the Obukhov length, here where y = r−r i . We realize that in the overlap region the viscous stresses are negligible so that uw ≈ u 2 τ and the gradient of the streamwise velocity in the shear dominated region is ∂ r U = uτ κy , see (1.1), which we take for reference in defining L c . We approximate the curvature production by U/r = ω i , and L c then becomes (4.6) We use κ = 0.39 throughout the paper, which is consistent with the data of Huisman et al. (2013), see figure 3, and also agrees with measurements of κ in turbulent BLs and turbulent channel flows (Marusic et al. 2010). However, we note that a range of κ are reported in literature (Smits et al. 2011), and the employed data here is not conclusive on the second decimal. A subtle difference with the definition of Bradshaw (1969) resides in the definition of the curvature production term. Bradshaw (1969) uses the wall normal production only (i.e. − 2 r uwU ), in strict analogy with the buoyancy production, that contains no streamwise production term. Here, however, we decide to use to sum of the streamwise and wall-normal curvature production terms (i.e. − 1 r uwU ) to account for the total effects of streamline curvature. 4.2. Development of the functional form of ω + (y + ) Figure 3(a) shows the angular velocity profiles for turbulent TC flow. For very high Re of O(10 6 ), we observe the existence of a logarithmic form of the angular velocity profile with κ ≈ 0.39 and B ≈ 5, in accordance with (4.1). However, the extent of the profile is very limited, namely 50 < y + < 600, as observed in Huisman et al. (2013), covering a much smaller spatial range than it would in canonical wall-turbulence systems (Pope 2000) at similar Re τ . Figure 3(b) presents the so-called diagnostic function, y + dω + dy + , which allows for a more detailed investigation of the slope of ω + (y + ). Even for these high Re flows, only a very small region of the profile coincides with the straight line with slope κ −1 , which in this representation represents the log-layer.
Following the analysis above, we expect the velocity profile to behave differently in the region where curvature effects play a role -in close analogy with the Monin-Obukhov similarity theory. Hence, we make the wall-normal distance dimensionless with L c , see (4.6). This is done in figure 4(b) where we plot the diagnostic function versus y/L c . The result is a near perfect collapse of the angular velocity profiles, directly justifying the use of L c in turbulent TC flow. In fact, the profiles not only collapse with respect to their wall-normal location, but also in terms of their vertical coordinate, i.e. the slope of ω + (y + ). This secondary flat regime with slope λ −1 exists for larger r > L c , than the κ −1 regime. We find that λ = 0.64.
From these observations in figure 4 we obtain the unknown function Φ ω ( y Lc ) in (1.3) for 0.20 < y/L c < 0.65: Consequently, we integrate dω + d(y/Lc) = 1 (y/Lc)λ and arrive at where K is an integration constant and log is the natural logarithm. The offset K of this second regime at larger r is related to the height at which the first logarithmic regime at smaller r peels off to the second log regime. We thus expect that K = κ −1 log L + c + C which results in, where C is a constant equal to 1.0 (obtained by fitting to the highest Taylor number data). In figure 4(a) we plot ω + versus y/L c and subtract K to highlight the collapse. Indeed, we observe a collapse of the profiles.

The constant angular momentum region in the bulk
In the previous section we discussed the shape of the mean streamwise velocity profile in the IC BL, culminating in a new functional form which includes the stratification length L c . However, to arrive at a Nu(Ta) relationship, we need to assess the velocity profile in the bulk region, too. Wendt (1933)  Here, we plot the constant angular momentum region in figure 4. We find that the transition from a λ −1 region into a constant angular momentum ω + = ω + i (1 − r 2 i /(2r 2 )) region occurs at y = L c . As such, the bulk region is entirely dominated by curvature effects of the streamlines. Consequently, the IC BL thickness δ i is equal to the curvature Obukhov length, δ i ≈ L c (and δ o = 2.5L c ). Recently, a very similar thickness of the BL was empirically found by Cheng et al. (2019).

The outer cylinder boundary layer
Analogous to the IC BL we can analyse the OC BL in the spirit of the Monin-Obukhov similarity theory. As mentioned in §4, Huisman et al. (2013) also obtained velocity profiles of the OC BL for the highest five Ta number experiments. From (4.5) we derive that  i /2, as derived by Townsend (1956), which very closely fits the data at y > L c . (b) Diagnostic function versus the rescaled wall normal distance y/L c = (r o −r)/L c , where L c = u τ,i /(κω i ) is the curvature Obukhov length. For higher r (r > r i + L c ) the shear dominated logarithmic regime with slope κ −1 peels off into a second logarithmic regime with slope λ −1 . The inset to (a) shows the mean angular velocity versus the wall normal distance y + = (r o −r)/δ ν,o . Data from the PIV measurements of Huisman et al. (2013). the relevant length scale for the OC BL is L c,o = ru τ,o /(κU ) with y = r o − r. We approximate the velocity scale U with ω i r i and the radius of curvature with r o , so that L c,o = u τ,i /(κω i ). The length scale is the same as L c,i . Figure 5(b) presents the gradient of the OC BL velocity profiles versus the dimensionless wall-distance y/L c . Again, we observe collapse of the profiles in both the vertical direction and the horizontal direction. In the range 0.20 < y/L c < 0.65 the gradient of the profiles is λ −1 , whose value is identical to the IC BL profiles. Since the findings in figure 5(b) are the same as in figure 4(b), we derive the velocity profile for the OC BL in the same manner as (4.7-4.9) and arrive at (4.10) where C o = 2.0 is obtained from fits in figure 5(a). Again, the profiles in figure 5(a) exhibit excellent overlap between (4.10) and the experimental data, especially at the highest two Ta numbers (see inset). We note that Re τ,o at the OC BL is smaller than Re τ,i at the IC BL, and consequently, we expect that the data at lower Ta still suffers from insufficient scale separation. We find that the obtained value for C in (4.9) differs from C o in (4.10). This is related to the different velocity scale in L c for the inner and outer cylinder BL. Once we estimate L c,o = u τ,i r i /(κU ), where U = 0.4ω i r i is the angular velocity scale in the outer BL as obtained from the data (T a = 6.1 × 10 12 ), the constant C 0 = 2 − (κ −1 − λ −1 ) log 0.4 = 1.1 ≈ C is consistent with (4.9).

The effects of the radius ratio η
Up to this point, we have shown that one can treat inner cylinder rotating TC flow as an unstably stratified turbulent shear flow, in close analogy with temperature stratified flows. We proposed a new functional form of the mean angular velocity in (4.9) that well describes the experimental profiles measured by Huisman et al. (2013) in both inner and outer BL for all Re at η = 0.716. The question arises what implications of the theory of stratified flows -and consequently (4.9) -bring to TC turbulence at varying radius ratios. To answer this question we first analyze DNS data of Ostilla-Mónico et al. (2015) and PIV data of van der Veen et al. (2016) at a lower radius ratio of η = 0.50 (corresponding to larger curvature effects), followed by the analysis of the DNS Ostilla-Mónico et al. (2015) data at a high radius ratio of η = 0.91. 5.1. Radius ratio η = 0.5 Figure 6 presents the velocity profiles at η = 0.5. The black solid line represents DNS data at a remarkable high Ta of 1.0 × 10 11 resulting in a significant scale separation; Re τ = 3257, see table 1. Nevertheless, the diagnostic function in figure 6(b) does not portray a shear dominated κ −1 regime, i.e. the solid black line never follows the black dotted line. However, at y/L c ≈ 0.20 the λ −1 regime is obtained. Note that we do not fit λ −1 to the data, but only use the value (λ = 0.64) as obtained in section 4. The dark grey solid line departs from the λ −1 region around y/L c ≈ 0.65, to follow the M o = ω i r 2 i /2 scaling of the bulk. This is in agreement with the observations at η = 0.716.
To understand the absence of a κ −1 region for this low η, we refer to the scale separation in table 1. A κ −1 slope requires that 1 y + 0.20L + c . However, for η = 0.50 at Ta = 1.0 × 10 11 we find that 0.20L + c ≈ 100, defying the existence of a shear production dominated region. The extensive scale separation between L + c and Re τ permits a large curvature dominated flow region where the angular momentum becomes constant, see figure 6(a). Figure 6 also presents in color the PIV data at low Ta. Although the scale  separation is generally very low, with Re τ not exceeding 10 3 , we find a trend towards the λ −1 region with increasing Ta. Especially figure 6(a) exhibits a collapse of the velocity profiles with (4.9) at higher Ta. 5.2. Radius ratio η = 0.909 Figure 7 shows data from a DNS at high η = 0.91 (corresponding to small curvature effects) and Ta = 1.0 × 10 11 . Interestingly, we observe a pronounced κ −1 region. However there is a total absence of the λ −1 and the M o region. Once again this is understood with the scale separation argument. In this case L + c > Re τ , and therefore there is no location in the flow where the curvature effects dominate, see table 1.

General radius ratio η
To close this section, we provide a phase diagram of the scale separation at Re τ ≈ 3000 for varying η, in order to illustrate where one would expect to see κ −1 , λ −1 , and constant angular momentum regions of the angular velocity profile, in figure 8. We base the phase diagram on three cases for η = (0.500, 0.716, 0.909) and Re τ ≈ 3000, for which we calculate the phase boundaries, see table 1. Note that the boundaries are not sharp, and gradual changes in the relative importance of TKE production by shear and curvature lead to new regions. However, we now immediately see from the diagram that for high η the Obukhov curvature BL is only expected to appear distinctly at extremely high Re τ (higher than Re τ = 3000). In contrast, for low η, we need extremely high Re τ (higher than Re τ = 3000) to observe the Prandtl-von Kármán turbulent BL type.

The Nu(Ta) and C f (Re i ) relationships
The derivation of the angular velocity profile in a turbulent BL with strong curvature effects, see (4.9), allows us to derive a functional form that relates the dimensionless torque Nu to the dimensionless driving Ta at Re o = 0. To do so, we follow the very recent work by Cheng et al. (2019). Therein, the BL profile (the conventional shear dominated von Kármán type) is matched with the constant angular momentum bulk profile at the edge of the BL. With a fitting constant for the BL thickness, Cheng et al. (2019) arrive at a very accurate prediction of Nu over a wide range of Ta. Here, we match the angular velocity profiles in the bulk and the BL at the BL height δ = αL c . Note that the constant α is easily extracted from figure 4, where it refers to the outer bound of the λ −1 region -where the BL and bulk meet.
(6.1) We realize that L c = 2r i Re τ /(κRe i ), L + c = (4ηRe 2 τ )/(κ(1 − η)Re i ), and ω i /ω τ,i = Re i /(2Re τ ) so that we can rewrite (6.1), where W (Z) is the principal branch of the Lambert W function. Figure 9 presents the prediction of equation (6.3) together with 8 datasets from DNS and experiments -covering 0.357 η 0.909 and 7 orders of magnitude in Ta. α = 0.65, see (4.7) and figure 4. Naturally, we find deviations at low Ta, where the BLs are not fully turbulent yet. However, we find good overlap at high Ta for various η. For high η, (6.3) looses its validity since shear is dominating curvature effects throughout the entire BL at the current Ta. The Nu(Ta) relation is thus better described by the functional form derived in Cheng et al. (2019). However, we note that the ratio Re τ /L + c will become The friction factor C f versus the the IC Reynolds number Re i . Colours and symbols are the same as in figure 9 and links to the references can be found in the caption of that figure. larger with increasing Ta, so that for extremely high Ta (even much higher than 10 12 ), the Nu(Ta) relationship at η = 0.909 will also follow (6.3). For Ta < 10 6 , the BLs are of the laminar type and Nu scales with Ta 1/3 (Ostilla-Mónico et al. 2014a). Figure 10(a) shows the Nu(Ta) relationship where Nu is compensated with Ta 1/3 , such that we highlight the transition to a turbulent BL where the scaling exponent is larger than 1/3. We emphasize that only after this transition, which is gradual and appears to depend on η, when BLs are entirely turbulent, equation (6.3) will correctly calculate Nu(Ta). Figure 10(b) presents the C f (Re i ) diagram, which is more conventionally used in the pipe flow and BL flow communities. The solid lines are given by equation (6.3) where the friction factor is calculated from C f = 4Nu/(η(1 + η)Re i ).

Summary and Conclusions
In summary, we have developed a theory, similar to that of thermally stratified turbulent BLs, as famously developed by Monin & Obukhov (1954), for the curved turbulent BLs in inner cylinder rotating TC flow. In this analogy, the destabilizing effects from curvature of the streamlines in inner cylinder rotating TC flow are similar to the destabilizing effects coming from unstable thermal stratification in the atmospheric BL.
We show that the curvature Obukhov length L c (Bradshaw 1969) separates the spatial regions that are dominated by shear and curvature effects. We find that for δ ν < y 0.20L c , the mean angular velocity profile in the BL is described by the classical shear profile, with the slope given by the von Kármán constant κ −1 = 0.39 −1 . In contrast, for 0.20L c y 0.65L c , where curvature effects are relevant, the slope of the angular velocity profile is λ −1 = 0.64 −1 . For y 0.65L c curvature effects dominate, and a region with constant angular momentum sets in. This theory is applied to -and found consistent with -PIV measurements and high-fidelity DNS data covering a wide range of radius ratios 0.50 η 0.909 and rotation rates 10 8 Ta 10 12 , and describes both the IC BL and the OC BL.
Building on these findings we derived a new functional form of the mean angular velocity profile in TC turbulence, with separate spatial regions where curvature and shear effects are respectively relevant. Upon matching (Cheng et al. 2019) this BL profile with the constant angular momentum profile in the bulk, at the edge of the BL, we obtain a Nu(Ta) (and C f (Re i )) relation that agrees well with various data sets at high Ta and varying η.
Future research might investigate the effects of stably stratified TC flow (i.e. outer cylinder rotation), or even mixed stratified TC flow (i.e. counter cylinder rotation) within the framework of the Monin-Obukhov similarity theory. However, so far, only velocity profiles with a scale separation up to Re τ ≈ 1200 are available for OC rotation (Ostilla-Mónico et al. 2016) to apply the theoretical analysis. Also, based on the newly derived velocity profile, it becomes necessary to reassess the fully rough asymptote for rough wall turbulent TC flow (Berghout et al. 2019).