Precise measurements of torque in von Karman swirling flow driven by a bladed disk

Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by counter-rotating bladed disks reveals an apparent breaking of the law of similarity. Potentially, such breakdown could arise from several possible factors, including dependence on dimensionless numbers other that $Re$ or velocity coupling to other fields such as temperature. However, careful redesign and calibration of the experiment showed that this unexpected result was due to background errorscaused by minute misalignments which lead to a noisy and irreproducible torque signal at low rotation speeds and prevented correct background subtraction normally ascribed to frictional losses. An important lesson to be learnt is that multiple minute misalignments can nonlinearly couple to the torque signal and provide a dc offset that cannot be removed by averaging. That offset can cause the observed divergence of the friction coefficient C_f from its constant value observed in the turbulent regime. To minimize the friction and misalignments, we significantly modified the experimental setup and carried out the experiment with one bladed disk where the disk, torque meter and motor shaft axes can be aligned with significantly smaller error, close to the torque meter resolution. As a result we made precise measurements with high resolution and sensitivity of the small torques produced for low rotation speeds for several water-glycerin solutions of different viscosities and confirmed the similarity law in a wide range of Re in particular in low viscosity fluids.


Introduction
The von Karman swirling flow between two rotating coaxial impellers has been extensively studied over the last two decades [1][2][3][4][5][6][7]. In this flow, disks of radius R counter-rotating with the same angular velocity (rad/s) are used to mechanically stir the fluid, thereby injecting energy in the bulk from the surface of the disks. Experiments have shown that attaching blades to the disk surface are more effective at injecting energy, compared to a smooth surface (known as viscous stirring), and the energy is directly supplied to the vortices trailing behind the blades and with a size comparable to the disk. Known as inertial stirring, such method of energy injection can create very high turbulence levels with velocity fluctuations up to 50% of the blade velocity [7]. The blades can be flat or curved, where in the latter case the different direction of curvature of the blades relative to the rotation can lead to different dynamics [7]. Although such a flow is far from the theoretically explored regime of homogenous, isotropic turbulence in an infinite space, it has been experimentally shown to reproduce the theoretically predicted scaling relation of injected and dissipated energy [2,7,8]. Besides, inertial stirring also sets a well-defined spatial scale for energy injection, almost an order of magnitude larger than that for viscous stirring [8,9].
A closed rotating flow, as described above, and its transition to turbulence can be quantitatively characterised by torque measurements. In such flows, the normalised average torque,¯ , the friction coefficient, C f =¯ /ρ 2 R 5 , should solely be a function of Re = ρ R 2 /η, according to predictions from the law of self-similarity, where ρ and η are the fluid density and dynamic viscosity, respectively. Earlier torque measurements performed in very high viscosity water-glycerine solutions [7] with glycerine concentrations from 74% up to 99% w/w and two temperatures 15 • C and 30 • C show that C f decreases from Re 50 till Re c 3.3 × 10 3 , where a transition from laminar to a fully developed turbulent regime takes place, and becomes constant valued, independent of Re. However, the data for C f of water-glycerine solutions in a laminar as well as transition region overlap within large error bars and do not provide the requisite accuracy for such measurements. Moreover, no data on lower viscosity solutions were reported, except for several points in the turbulent regime in water at Re ≥ 10 6 [6,7].
More detailed data on C f in the turbulent regime taken solely for water-sugar solutions at different sugar concentrations from 0% up to 40% at a constant temperature of 24 ± 0.1 • C corresponding to the low dynamic viscosities η from about 1 up to 6.1 mPa · s were presented in Ref. [8], where only the constant values of C f in a wide range of Re are shown.
Contrary to the data reviewed above, torque measurements in water-sugar solutions at constant temperature conditions by a torque meter of much higher resolution and accuracy than those used in all former experiments showed a surprising deviation of C f from the constant value for each solution at low η and and corresponding value of Re, supposedly in the fully turbulent regime and depended on the solution viscosity η [10]. Hence the data show that Re c at the laminar-to-turbulent transition for solutions of different sugar concentrations depends explicitly on the solution viscosity meaning that C f changes due to viscosity variations even if the value of Re is kept invariant. Indeed, in the range of Re between ∼ 5 × 10 4 and 10 6 several values of C f for a single value of Re were found [10]. From both theoretical (dimensional reasoning of the Navier-Stokes equations for fluid flow) and experimental perspectives (measurements of pipe flow resistance), we expect self-similarity; C f must depend only on Re and not explicitly on the viscosity. Hence, it is of interest to understand the observed violation as it implies either new or unaccounted physics or imprecision in the experimental apparatus.
To proceed we first put forth three hypotheses that can explain the violation and analyse each one in turn. First, data for the torque may not have been collected and thus not averaged for a sufficiently long enough time scale to assure asymptotically stationary statistics such that the mean torque is independent of the averaging time. Second, violation may occur if the fluid velocity field interacts with another field, the most obvious of which is a temperature field. In the turbulent state, giant velocity fluctuations can lead to a non-uniform temperature distribution due to non-uniform dissipation and generate heat fluxes. Thus, it is possible that Re is not the only dimensionless control parameter in the system, and that others like the Prandtl number Pr = η/ρκ and the Rayleigh number Ra = Tl 3 gαρ/ηκ = Tl 3 gαρ 2 Pr/η 2 couple the fluid flow velocity to heat fluxes. Here α and κ are fluid thermal diffusivity and thermal expansion coefficients, respectively, g is the gravitational constant, and T is the temperature difference on the spatial scale l. A third possible reason is the low resolution of the torque measurements due to large irreproducible and unpredictable background errors at the level of the torque signal despite the fact that our torque measurements are more precise than those reported in all previous experiments in a swirling flow (see [8]). Thus, the main goal of these studies is precise instrumentation; to minimise background errors below that of the small torque signals thereby improving upon previously achieved signal to noise ratios at very small signal values and measure the friction coefficient as a function of Re with accuracy of the torque meter resolution.
The paper is organised as follows. We first present the measurements of torque, pressure fluctuations, and flow velocity in the swirling flow of water-glycerine solutions in a wide range of viscosities driven by two counter-rotating, curved bladed disks in the experimental setup similar to that already reported in Ref. [8]. In these measurements a sequence of laminar to turbulent transitions were observed with corresponding critical Re depending on the fluid viscosity. We also describe pressure fluctuations and local two-component velocity measurements simultaneously carried out in a wide range of Re which do not provide any evidence of the laminar-turbulent transition, except for the one known to occur at Re ≈ 3.3 × 10 3 . Then we report the detailed precise torque measurements with even a more sensitive torque meter first in a swirling flow driven by two counter-rotating bladed disks at low which are followed by the precise torque measurements in a swirling flow driven by a single bladed disk of the same solutions in a modified setup. The latter is designed to precisely align the upper disk, the torque meter, and the motor axes with results in the ability to conduct the torque measurements with almost two orders of magnitude higher resolution and accuracy than in our previous experiments. A detailed discussion of background errors due to friction losses but mostly due to misalignments is provided with estimates and experimental proof that the fine alignment leads to the expected self-similarity within the torque meter resolution.

Experimental setup
In these studies we used two different setups.
The first setup is the same as used in previous experiments and reported in Ref. [8]. It consists of a cylindrical vessel where the swirling flow is generated made of Lucite (Plexiglas) of diameter D = 29 cm and height L = 31 cm surrounded by a hexagon shaped Lucite enclosure with a gap between them filled with water for thermal regulation. Two 10-mm thick Lucite disks of R = 13.2 cm and H = 19 cm apart were used as impellers and had 8 curved blades of 10 mm height attached to it with radius of curvature 7 cm corresponding to a blade inclination angle of α = 70.5 • (see Figure 1). The stirrers rotate with the concave face of the blades pushing the fluid driven by two brushless sinusoidal motors F-6100 (Electro-Craft servo systems) with maximum continuous torque of 13 Nm and peak torque 31.1 Nm controlled by a motion card PCI-7344NI (Advanced Motion Controls) via optical encoders with constant velocity within 0.1% up to /2π = 9 Hz (540 rpm) limited by the overheating of the motion control card. Two different seals were used to prevent water leakage from penetrating into the ball-bearings: a rotating SS-R00 John crane type R00, AES P08, wave spring design, balanced end-face mechanical seal with guarantied limited mechanical friction at the bottom motor driving shaft, and an oil seal CR14223 with low mechanical friction from CR Services Co at the top motor driving shaft.
The measured average torque¯ 0 was corrected for background errors that consists of friction losses and other systematic sources of errors¯ err obtained from the torque measurements in air with disks with curved blades yielding the following expression =¯ 0 −¯ err . Estimates show that the moment of inertia of water is about 50 times larger than the total moment of inertia of the rotor, the bladed stirrer, and the torque meter resulting in a cut-off frequency of the control of about 20 Hz. Thus, in constant angular velocity mode with the feedback loop control the torque spectral measurements are reliable up to about 10 Hz. The temperature of the fluid was measured by three thermistors placed at three different heights along the side of the container at 2, 20, and 28 cm, respectively, measured from the top lid using a RTD thermometer for calibration. The temperature was stabilised within ±0.1 • C by three circulating refrigerators (Lauda Inc.) driving cooling water via coiled copper tubes soldered to the top and bottom cylinder lids made of stainless steel and via the space between the hexagon shaped Lucite enclosure and the cylindrical vessel.
The torque required to generate the swirling flow was measured using a torque meter inserted between the upper disk and the motor (see Figure 1) when the disks rotate in opposite directions at equal rates, by a non-contact calibrated torque meter MCRT 49001 (S. Himmelstein Co.) within the range up to 11.3 Nm (with overload up to 45.2 Nm) with 0.1% accuracy and up to 15,000 rpm. The torque meter has two low-pass filter outputs: one with bandwidth between dc and 500 Hz, which we used in the measurements, and another one with bandwidth between dc and 1 Hz used to verify the average value of torque with output rms noise 0.1% and 0.01% of the full scale, respectively. For a second, more precise measurement using the same setup, another torque meter MCRT no. 48999VB(5-1)NFNN (S. Himmelstein Co.) with a range up to 0.35 Nm and overload up to 0.7 Nm with 0.1% resolution was used. The torque was measured at constant rotation rates of the disks stabilised through feedback control and optical encoder and used to test the stability of the measurements via deviation of from constant value due to temperature variations or other factors.
The pressure fluctuations were measured by a 40PC001B miniature signal conditioned pressure sensor (Honeywell Co.) with 2.8 mm diameter membrane and working pressure range up to ±6.7 kPa with accuracy up to 0.2%, resolution 0.3 mV/Pa and rise time less than 1 msec. However, the limited working pressure range of this pressure sensor did not allow us to use it at sufficiently high Re. Both torque and pressure data were acquired with a sampling rate of about 83 Hz to avoid high-frequency noise, more than 4 times larger than the cut-off frequency mentioned above. Since we were not interested in studying the frequency power spectra of pressure in an inertial range but just to observe the transition to turbulence, the frequency range below 100 Hz was sufficient to meet our goals.
In the first experimental setup, three different experiments with water-glycerine solutions of different concentrations, temperatures, and so different dynamic viscosities η as working Newtonian fluids were conducted. The values of these parameters of the corresponding water-glycerine solutions are presented in Tables Table 2.
The second experimental setup consisted of the same apparatus to generate a swirling flow, however only an upper bladed disk was used driven by the same brushless DC motor as previously used. In this case, the rotating disk was separated from the cylinder bottom considered as a stationary disk by a distance H = 24 cm (R/H = 0.55) (see Figure 2). The torque applied on the upper disk to generate the swirling flow was measured by the same non-contact calibrated, more precise torque meter MCRT no. 48999VB(5-1)NFNN (S. Himmelstein Co.) mentioned above.
In order to conduct precise measurements at low frequencies and extend the range of the torque meter, both the friction and misalignment had to be minimised. High precision machining and low friction ball-bearings helped alleviate the first problem. To minimise the errors of the second kind the following modifications were made. Traditionally, soft helical spring couplers have been used as a solution to slight misalignment between two axes. Attaching spring couplers in our system caused the measured torque to oscillate at particular drive frequencies due to unavoidable nonlinear resonances. For accurate calibration, the torque meter rotation axis was precisely aligned to that of the disk shaft by attaching the former to a two-axis rotation stage attached to a XY translation stage. This configuration allowed enough degrees of freedom to align both the position and the angle of the torque meter. The radius of the disk shaft was 12.5 mm while that of the torque meter was 3.15 mm. The coupler from the disk to torque meter was machined out of aluminium and was tapered down, being wide (31 mm) at the bottom and narrow at the top (9.5 mm). This coupler design lowered its centre of mass considerably and reduced nutation errors. At the top of the coupler a narrow notch (width 8 mm) was machined, into which the corresponding protrusion from the torque meter fit in precisely, analogous to a flat head screw driver and screw. This arrangement minimised bending torques between the torque meter and the disk shaft but caused a small mismatch in the shaft angles between the torque meter and the DC motor. The centres of the motor and torque meter were aligned by placing the former on a XY translation stage while errors due to angle misalignment was minimised by connecting the motor and torque meter with a two-arm connector which allowed two extra degrees of freedom. After these modifications, the background DC signal was stable to within 0.2 mNm, close to the precision allowed by the instrument. We also observed that at a low rotation speed the background torque does indeed increase due to a persistent misalignment owing to a nutation of the disk shaft coupler (this occurs because the shaft is not perfectly aligned with gravity). In the second experimental setup, water-glycerine solutions of different concentrations, the same temperature and so different dynamic viscosities η as working Newtonian fluids were used. The values of these parameters are presented in Table 4.

Large range torque meter
For the first data set, we obtained average torque¯ 0 as a function of the rotation frequency f (f = 60 /2π) with the large range torque meter in a wide range of f from 1 up to 540 rpm and η from 0.6 up to 1432 mPa · s for water-glycerine solutions from zero up to 99% w/w of glycerine and various temperatures (see Figure 1SM and Table 1). The data have a ∼ f 2 -dependence which corroborates existing knowledge that in a turbulent regime the friction factor C f is independent of Re. However, on a smaller frequency scale,¯ 0 shows an abnormal dependence on frequency and large scatter, especially at f < 30 rpm in water, (see Figures 1SM, 2SMa,b in Supplementary Material) despite the fact that torque meter resolution and accuracy is about 30 times higher than previously reported measurements (see insets in Figures 1SM, 2SMa,b). Long time series data collected of of a solution with 60% w/w glycerine at 24 • C, (Figure 3SMa) show rare and strong bursts, much larger than the measurement resolution, and may evidence unexpected non-stationary statistics, but data collected at higher Re (Figure 3SMb) have much smaller scatter of about ±20 mNm in and a stable average value (see also Figure 4SM, at higher values of Re for the solution containing 20% w/w glycerine at 21 • C). In addition to the problem mentioned above, background errors err measured in the absence of fluid, necessary to compute the friction coefficient, are noisy and irreproducible especially at low f. Instead of subtracting err , to compute C f , we subtracted from the raw torque measurements the offset of¯ 0 in the limit f → 0. The results (see Figure 5SM) are in fair agreement with the results published in Ref. [7], except that for low viscosity solutions our measurements show an anomalous divergence of C f not only as a function of Re but also as a function of η, an unexpected result that needs careful consideration (the data are not shown).
To avoid the anomalous region f < 30 rpm, the next set of experiments were conducted with the same set up but in the range ≥ 3 rad/s (or at f ≥ 28.7 rpm), where the data for 0 are smooth with negligible scatter (see Figure 3(a)) but still have the offset from zero as → 0 (inset of Figure 3(a)). As in our former measurements, err reaches large values up to 0.3 Nm and is irreproducible at the level of ∼ 50 mNm, as shown in the inset in Figure 3(a). Hence, to compute C f as a function of Re, we shifted the average torque in   Figure 3(c)) causes all of the C f data to collapse onto a single curve (see Figure 3(c)). We should note that H is a phenomenological parameter and not unique; any function aH, where a is a constant, also causes all the data to collapse. This analysis shows that low f behaviour of the system must be carefully analysed, and will be discussed in the next section. Concurrently, rms /¯ for water-glycerine solutions in a whole range of viscosities show strong divergence from almost Re independent behaviour at each η (see Figure 3(d)). This is however not a surprising result, since we have already shown that the measurement for¯ at low f may be unreliable. To summarise, we improved the resolution of our experiment to a higher value from previous measurements using a high-resolution torque meter and analysed the data only above some rotation frequency but observed that the apparent violation of the similarity law persists.
To check if similar divergences occurred in other parameters of the system, we simultaneously measured p rms (in the first run), shown in its scaled form as C p = p rms /ρ 2 R 2 versus Re in Figure 4, and of two components of flow velocity and their fluctuations shown in Figure 5. From Figure 4, we find that C p is nearly independent of Re between Re ≥ 2 × 10 4 and 7 × 10 5 with small scatter at lower values of Re that arises from the lack of resolution of the pressure transducer at low signal values. The measurements of the azimuthalV x ,V y and axial V rms x , V rms y flow velocity components versus Re for four water-glycerine solutions ( Figure 5) reveal several transitions, from which we marked only two by arrows: at Re ≈ 330 and Re c ≈ 3300. The first one (marked by arrows in Figure 5(d)) defined by an increase of bothV y and V rms y is the first bifurcation in the laminar flow field, whereas the second transition Re c (marked by arrows in Figure 5(c)), which is directly related to the subject of the paper, indicates the onset of turbulence. For both bifurcations, the smooth continuous transitions of both velocity components are found in the average velocities as well as their rms fluctuations, though one notices that the axial x , V rms y on Re for four water-glycerine solutions for the first setup: (a) 30% w/w glycerine at 15 • C; (b) 60% w/w glycerine at 17 • C; (c) 93% w/w glycerine at 30 • C; arrows indicate the laminar-to-turbulence transition at Re c ≈ 3300 for both V y and V rms y ; (d) 93% w/w glycerine at 15 • C; arrows indicate the first bifurcation in a laminar flow at Re ≈ 330 for both V y and V rms y .
velocity component shows both smooth transitions more clearly inV y as well as V rms y . Moreover, at Re > 3300 up to the highest values, smooth behaviour is found in a contrast to that in the torque data. Both of these transitions were reported early elsewhere (see for example [7]).
In conclusion, using the large torque meter we demonstrated that at > 3 rad/s, the torque statistics is stationary, and a time series for about one hour is sufficient to characterise its average and rms fluctuation values. Hence, the first of our tests proved to be negative. To test the presence of additional control parameters, such as Pr and Ra, we used different proportions of water-glycerine solutions with the same viscosities, and hence same Re, but different Pr and Ra. Then a comparison of the measurements on¯ versus Re for both solutions could reveal the influence, if any, of these other parameters of C f . Unfortunately, since these measurements should be performed at low , the torque measurements with the large torque meter do not have sufficient resolution and accuracy to reveal differences if any and in the next section more accurate measurements will be discussed.
Since the divergences of C f occurred at low rotation speeds, we conjectured that it arose due to limited device resolution. Thus, we needed to increase signal-to-noise ratio at low rotation speeds to better resolve the signal. To achieve this, we used a higher resolution torque meter, which allowed us to better analyse the signal and find the source of the noise. Moreover, errors with the small torque meter were also reduced, since a smaller torque meter required less bulky couplers that were difficult to align. A switch to smaller couplers inherently entails better alignment due to geometrical restrictions.

Small range torque meter
To further increase the resolution of the torque measurements and discover the source of irreproducibility in err a much more sensitive and higher resolution torque meter was used in the same experimental setup. Installing this newer, physically smaller and lighter torque meter lowered the offset and irreproducibility of the measurements, but did not eliminate them. Instead of 0.3 Nm, the offset was reduced to 5 mNm and irreproducibility brought down to a level of 10 mNm (inset of Figure 6(b)), but nevertheless 30 times larger than the torque meter resolution. Such drastic reduction allowed us to reliably measure smaller torques, but just a subtraction of err from the average torque to remove the offset is in principle not the correct procedure. Figure 6(a) shows the dependence of¯ corrected by a shift of 5 mNm in a wide range of water-glycerine solution viscosities and starting from ∼ 1.2 rad/s (or f 11.46 rpm). Due to smaller range of the torque meter, only glycerol-water mixtures with viscosity up to 18.86 mPa · s were used in these experiments. The data are smooth with very small scatter even on the plot with larger resolution shown in the inset in Figure 6(a). The resulting friction coefficient is constant in a wide range of Re values from about 2 × 10 5 down to Re c ≈ 3 × 10 3 (see Figure 6(b)), though the constant value of C f is smaller due to new low friction ball-bearings and seals used. The dependence of rms /¯ on Re is presented in Figure 6(c).
The torque resolution is sufficient to compare different pairs of water-glycerine solutions with the same viscosities pairwise but different Pr and Ra. The comparison for the first pair of water at 10.5 • C and a water-glycerine solution of 30% w/w glycerine at 44 • C with the same η = 1.3 mPa · s, close values of Pr w = 9.63 and Pr Gl30 = 9.53, respectively, and the ratio Ra w /Ra Gl30 ≈ 7 reveals a significant difference in C f at smaller values of Re (see Figure 7(a)). Contrarily, the comparison for another pair of water-glycerine solutions of 10% w/w glycerine at 10 • C and 40% w/w glycerine at 45 • C with almost the same η = 1.75 mPa · s and Pr 12.3 but different Ra with their ratio Ra Gl40 /Ra GL10 ≈ 4, does not show any significant difference (see Figure 7(b)). Since we used a zero frequency offset shift in¯ 0 to compute C f instead of subtracting irreproducible and noisy err , which is the likely source of the contradicting results for two pairs of solutions, the significance of Ra as an additional parameter is doubtful and unlikely, and we claim that second possible source of the validation of the similarity law gives a negative result: there is only one control parameter in the flow, namely Re.
Even though we significantly reduced err , it remained irreproducible and about 30 times larger than the precise torque meter resolution. We obtained Re-independent friction factor in the wide range of Re seemingly resolving the breaking of self-similarity puzzle but using a shift to zero of¯ 0 , an unfounded approach, instead of subtracting err . The sensitive torque meter allowed us a separate insight; the presence of slight misalignments of the torque meter rotation axis with the upper disk and motor shaft and a separate misalignment of the lower motor shaft can nonlinearly couple to produce a dc offset in the torque signal. The irreproducibility arose from minute mechanical movements of the independent parts of the apparatus due to the swirling of a large body of fluid, that changed the 'phases' of the individual components relative to one another and changed the dc offset.
Since it was impossible to sufficiently precise align the upper and lower drives due to the existing design, we switched to a one-disk arrangement in the second setup.

Torque error estimates and experimental details
As we concluded above, the observed anomalous divergence of C f from its constant value at Re, which depends solely on η, is explained by the presence of irreproducible and noisy err , which consists of both frictional losses and misalignment errors, though irreproducibility results mostly from misalignments. To understand the cause of this observed divergence and divulge the unknown error mechanism, we switched to a more sensitive torque meter with 0.35 Nm range with a precision of 0.1% or 0.35 mNm, enough to resolve the small torques at low rotation speeds. However, even with these sensitive torque measurements the divergences in C f persisted. In this case, the total measured torque¯ 0 includes two parts: the signal sig ≡¯ and the background errors err , which consist of friction from the ball-bearings and seals and misalignments. Then the total friction coefficient is The expression for C f can be rewritten via Re as C f = (¯ + err )ρ/Re 2 η 2 R. In the fully turbulent state C f is a constant independent of Re as shown above and in Refs. [7,8]. From the expression for C f one finds that, if err ¯ , then one gets C f ∼ 1/Re 2 . Hence the Re, at which the background errors become significant, is given by the expression Re t ≈ err ρ/Cη 2 R, where C is the constant value of C f in the turbulent regime. Substituting err with the value of the measured background, for example, in the case of the small precise torque meter discussed at the end of the previous subsection, one gets a value of Re t , at which C f deviates from the constant value. Then for err ≈ 10 mNm (see the inset in Figure 6(b)) one gets for water at T = 10 • C (see Table 3) Re t = 3.1 × 10 4 that agrees rather well with the lowest Re observed for water in the experiment presented in the inset and the main plot in Figure 6(b). However, the deviation in C f from the constant value still does not show up there, since Re t corresponds to 2 rad/s, below which the measurements were not extended due to larger scatter. Similar accordance is found for the rest of data shown in Figure 6(a,b) and at each η the measurements were ceased for between 1.2 and 2 rad/s, where the scatter is enhanced and C f diverges. On the other hand, in the case of measurements with the large torque meter, err was an order of magnitude or more larger and divergence of C f from the constant value occurred at > 3 rad/s, at which the measurements were still conducted. For example, for water at T = 15 • C we obtain Re t 9.5 × 10 4 a factor 3-4 smaller than the value of Re observed in the experiment, at which C f starts to deviate from the constant (see Figure 3(b)). The value of Re t corresponds to 5.5 rad/s, at which¯ 0 0.16 Nm and C f exceeds the constant by ∼ 15%. It means that the real value of err is 15% of the signal¯ , and its contribution to C f grows with a decreasing rotation speed as −2 . Thus, the C f divergence measured with the large torque meter can be attributed to either an unjustified shift to zero of the measured¯ 0 or an incorrect subtraction of the background signal, which is stable and approximately constant for high rotation speeds but noisy and irreproducible at low rotation speeds.

Precise measurements of torque in a swirling flow driven by a single disk
In spite of a reduction in err achieved in the one-disk configuration (see inset in Figure 8) with careful measurements using the small precise torque meter, it is still a factor of 20 larger than the resolution of the device, 0.35 mNm. The measured background is also irreproducible and noisy preventing correct calibration of the device for < 10 rad/s. Since a large volume of a working fluid is driven by a large and powerful motor, it is not surprising that a slight misalignments between the motor, torque meter, and disk axes strongly affects the torque measurements. Due to the misalignment the shaft attached to the spinning disk undergoes a slight nutation, which is exaggerated by the height of couplers used to attach to the torque meter. The misalignment of the torque meter rotation axis and the motor shaft and the nutation of the disk shaft coupler causes small periodic variations in the measured torque, which if occurring singly can be filtered out from the measured signal. But both these error mechanisms exist and in unison couple nonlinearly to produce a DC offset in the measured torque and cannot be filtered out. Hence, the background signal measured is a sum of the friction in the bearings and seals that is small in comparison and the DC offset caused by the misalignment, and err consists of terms due to both the frictional losses and misalignment errors.
In order to perform precise torque measurements at low frequencies and extend the range of the torque meter, the background errors were minimised by an alignment of the torque meter rotation axis with the disk and motor shafts, as it described in Section 2. First, time series of the torque measurements at a constant was used to test the stability of the measurements from the deviation of¯ from its constant value. The degree of long-term stability of¯ 0 for the glycerine-water solutions with c = 50% w/w of glycerine at 20.2 • C and Re = 10 4 and c = 85% w/w of glycerine at 20.2 • C and Re = 10 3 can be estimated from the data presented in Figure 6SM in Supplementary Material, where only a part of the data is shown. Stationarity of statistics is demonstrated by Gaussian probability distribution functions in Figure 7SM and flat torque power spectra in Figure 8SM in Supplementary Material for both solutions.
After careful alignment background errors were reduced to ∼ 3 mNm at > 6 rad/s, about 8.5 times larger than the device resolution and almost a factor 30 smaller than that measured with the large torque meter (see Figure 8). In the inset in Figure 8, err is shown before alignment, and although the scatter is small and still irreproducible. The value of the  The average torque 0 as a function of angular velocity in a wide range of its variations down to 1.5 rad/s for water-glycerine solutions in a wide range of w/w glycerine concentrations at temperature T = 24 • C for the second setup with a single rotating bladed disk and a high precision small torque meter. Inset: the same data at higher resolution at low . (b) Friction coefficient C f versus Re for the same data. Inset: turbulent intensity rms / versus Re for the same data.
background decreases from = 2 rad/s to = 6 rad/s by a factor of 1.6 but in the whole range of it is reproducible for different experimental runs and shows small scatter in Figure 8. After these modifications, the background DC signal was stable and reproducible to within 0.2 mNm, close to the precision allowed by the instrument. Thus, after the correct background subtraction and filtering the torque signal first by a 2 Hz low-pass filter to remove the PID signal and second by a notch filter to remove the drive frequency, the measured torque is precise with 1 mNm due to any misalignment errors. Instead of using the zero offset subtraction procedure to calculate C f , we can simply subtract err from , since the errors are now small and reproducible (see the main plot and inset in Figure 9(a) with correct background subtraction 0 → 0 as → 0). Then C f obtained by subtracting err is shown in Figure 9(b) is free from any divergence. However, even after the careful alignment and measurement of the background torque, small deviations from the scaling dependence were observed for low rotation speeds. This stems from the fact that the background signal, specifically the nonlinear interaction between the disk shaft and motor shaft is slightly modified in presence of fluid (instead of air), which value is impossible to estimate. By reducing alignment errors we, first, are able to cut err down to the resolution of the torque meter, and, second, to get the value of Re at which the deviation from the self-similarity occurs, which is at Re ∼ 10 4 for water at T = 10 • C by our estimates. The dependence of rms /¯ on Re is presented in the inset in Figure 9(b) and is almost free from divergence observed with the large torque meter. With the precisely aligned torque meter not only can the torque fluctuations be measured precisely, but the rescaled torque fluctuations can also be seen to scale weakly, but in a self-similar manner with Re.

Conclusion
Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by the counter-rotating bladed disks revealed an apparent breaking of the law of similarity that could arise from several possible factors including dependence on dimensional numbers other that Re or velocity coupling to other fields such as temperature. However, careful redesign and calibration of the experiment showed this unexpected result was due to background errors caused by minute misalignments which lead to a noisy and irreproducible torque signal at low rotation speeds and prevented correct background subtraction normally ascribed to frictional losses. An important lesson to be learnt is that multiple minute misalignments can nonlinearly couple to the torque signal and provide a dc offset that cannot be removed by averaging and cause the observed divergence of the friction coefficient C f from its constant value observed in the turbulent regime. To minimise the friction and misalignments, we significantly modified the experimental setup and carried out the experiment with one bladed disk where the disk, torque meter, and motor shaft axes can be aligned with significantly smaller error, close to the torque meter resolution. As a result, we made precise measurements with high resolution and sensitivity of the small torques produced for low rotation speeds for several water-glycerine solutions of different viscosities and confirmed the similarity law in a wide range of Re in particular for low viscosity fluids.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was partially supported by The Israel Science Foundation (ISF) [grant number 882/15].